CA Algebra I: Factoring quadratics

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Kind: captions
Language: en

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All right, we're
on problem 44.

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And they said, which is the
factored form of 3a squared

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minus 24ab plus 48b squared?

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And if this confuses you with
a's and b's instead of an x

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there, I just like to think of
in this case, this looks like

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an x squared.

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So I like to think of it the
same way I would think of it

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in terms of if this was an x
squared minus some number

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times x plus some
other number.

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So let's put that in the
back of your head.

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If that's confuses you,
don't worry about it.

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But the first thing I like to
do is try to get rid of the

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coefficient on the, in this
case, the a squared term.

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Let's see if we can factor
out a number.

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Well, sure.

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All of these are
divisible by 3.

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3 is divisible by 3, 24 is
divisible by 3 and 48 is

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divisible by 3.

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So we should be able to factor
out a 3, so let's do that.

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So that is equal to 3 times
a squared minus 8ab.

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24 divided by 3 is
8 and there's a

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minus in front of it.

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Plus 48 divided by-- 3.

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My brain was getting
ahead of me.

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48 divided by 3 is
16b squared.

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Now to factor this we have to
think about it as, are there

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two numbers-- well,
think of it.

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Let me rewrite this
a little bit.

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If we view a as kind of the
independent variable-- you

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wouldn't have to do it, but I
just wanted you to visualize

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it properly.

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3a squared minus 8b times
a plus 16b squared.

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So if we viewed a squared as
kind of the independent

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variable or the x term, so now
this kind of has the shape of

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polynomials that hopefully
you're used to factoring a

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little bit.

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We just have to think, OK, are
there two numbers that add up

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to minus 8b, and that when I
multiply the two numbers, I

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get 16b squared?

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So first all, the number
is going to be

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in terms of b, obviously.

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Because if I'm adding them and
I get a b and-- I get b's.

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And if I square them
I get a b squared.

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And it also makes me lead to
it's the same number, just

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used twice.

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Otherwise, it would be weird
to get a squared here.

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So the number that should pop
out, you say, OK, what two

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numbers add up to minus 8
and that when I square

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it is equal to 16?

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Minus 4, right?

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Minus 4, minus 4 is minus 8.

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Minus 4 squared is 16.

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So the number we're talking
about is minus 4b, right?

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Minus 4b.

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b, that's a b.

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Plus minus 4b is equal
to minus 8b.

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And minus 4b squared is
equal to 16b squared.

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So we can factor this out to be
3-- let me switch colors.

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3 times.

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We said minus 4b
is the number.

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When you add it twice
you get minus 8b.

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And you square it,
you get that.

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So it's a minus 4b
times a minus 4b.

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And that is choice C.

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Next problem, 40--
did I skip 45?

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I did.

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Let me copy and paste
45 in here.

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Can't

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skip problems. 45.

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Copy and paste it in.

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OK.

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Maybe over do on top
of this one.

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Let me see if I can
erase this.

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Look at that.

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All right, now I can paste
45 and we're ready to go.

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Which is a factor of x squared
minus 11x plus 24?

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So there's two ways to do it.

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You could just factor this.

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Well, that's the easiest way.

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Or you could test for 0's.

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But we'll just factor.

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So if we wanted to say-- well,
let me just-- x squared minus

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11x plus 24.

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So just like the last problem
you have to say, what two

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numbers when I add them equal
to minus 11, and when I

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multiply them, is equal
to positive 24?

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So first of all, if I'm
multiplying them and I get a

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positive 24, they're either
both positive or

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they're both negative.

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Since this is a negative number
it tells me that they

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both have-- I mean they both
can't be positive.

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You can't add two positive
numbers and

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end up with a negative.

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So we're going to deal with
two negative numbers.

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So essentially, what two
negative numbers when I add

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them equals minus 11 and when I
multiplying them equals 24?

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And hopefully, 8 and
3 pop out at you.

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Because minus 8 plus minus
3 is equal to minus 11.

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That's that.

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And then minus 8 times minus
3 is positive 24.

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So this can be factored as x
minus 8 times x minus 3.

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And if we look at their
choices, they

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had x minus 3 there.

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And of course, don't
get this confused.

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If they said, solve this
equation, right?

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This is just an expression.

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Now it becomes an equation
if we put an equal sign.

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Then we would have this equaling
0 and then you could

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say the roots of this polynomial
or the solutions to

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the equation are what
makes this true.

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And then the solutions
wouldn't be

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minus 8 and minus 3.

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The solutions would be
what makes this 0.

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So it's be x is equal to 8 would
make this 0, or x is

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equal to 3 would make that 0.

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But anyway, I'm going off on a
tangent and that's not what

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they asked you.

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But I think that confuses
people sometimes.

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OK, 46.

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Which of the following shows
9t squared plus 12t plus 4

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factored completely?

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OK, now this is an interesting
one because immediately, when

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I look at the numbers, there's
not one number that I can just

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factor out of everything.

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9, 12, and 4, they don't have
any common factors, so I can't

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just do that simplification.

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So we're going to do a little
bit more complexity.

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But the best way to think about
is whatever's on the t

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squared, this is probably kind
of-- this whole expression, if

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we're trying to factor it into
two binomials or into one

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binomial, this is going to
be the first term of that

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binomial squared.

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So we're going to be dealing
with something like 3t.

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I'm just taking the square
root of 9t squared.

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3t plus some number.

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Let's say plus a.

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Times 3t plus b.

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And now we can actually
just multiply this out

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and see what happens.

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Well first of all, they gave
us a multiple choice.

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We could just multiply these
out and see what happens.

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But let's pretend like that they
didn't give us choices

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and we had to do this
in a vacuum.

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If we did this in a vacuum,
we would have

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to factor it ourselves.

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We wouldn't just be able
to test their choices.

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Let's do that.

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So if we multiply this
out, we have 3t

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times 3t is 9t squared.

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Then you 3t times b,
so it's plus 3bt.

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Plus 3at plus ab.

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So this simplifies to
9t squared plus.

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Now what are we adding?

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Let's see.

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It says 3 times, a plus b, t.

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So it'd be 3 times,
a plus b, t.

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I just added these two terms and
I factored out the t and

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the 3 plus ab.

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Well now we can do a little
pattern matching, right?

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We could say a plus b times
3 is equal to 12.

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So a plus b is equal to what?
a plus b is equal to 12.

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This whole coefficient right
here is a 12 up here.

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So a plus b, it must
be equal to 4.

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Because 4 times 3 is equal to
12. a plus b is equal to 4 and

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we have a times b is
also equal to 4.

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So the only number that I can
think of when I add them I get

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4, when I multiply them
I get 4 is 2 and 2.

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Both of these are 2.

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So if I were to factor this
completely I get 3t plus 2

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squared, essentially.

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Because both of these terms
are the same thing, and

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that's choice A.

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A faster way frankly, to do this
might have just been to

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multiply this out and say,
that's the same thing.

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Anyway, next problem.

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What is the complete
factorization

00:08:24.620 --> 00:08:28.400
of 32 minus 8z squared?

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So let's think about
this a little bit.

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So the first thing that I like
to do once again is to try to

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factor out any numbers that are
just common to all of the

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terms. So let me do that.

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So 32 minus 8z squared.

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8 goes into both of
these, right?

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So let's factor out an 8.

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And actually, let's factor out
a negative 8, and I'll show

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you why did that.

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Because I like to put the
z squared, I like

00:08:49.640 --> 00:08:50.970
that one to be positive.

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So let's factor out
a negative 8.

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And so you get a minus 8.

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And you didn't have
to do that.

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You could have just
factored an 8.

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What's 32 divided by minus 8?

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That's minus 4.

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And then minus 8z squared
divided by minus 8 is just

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plus z squared.

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And so we can rearrange this, so
this becomes minus 8, times

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z squared minus 4.

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And now I'll review this because
this is an Algebra 1

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test, so maybe this isn't
obvious to you.

00:09:22.360 --> 00:09:26.870
But in general, if you see
something like this, a plus b

00:09:26.870 --> 00:09:30.640
times a minus b-- and this might
be a good exercise for

00:09:30.640 --> 00:09:31.990
you to multiply this out.

00:09:31.990 --> 00:09:36.420
But this is equal to a squared
minus b squared, because the

00:09:36.420 --> 00:09:37.640
middle terms cancel out.

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And that's a good exercise
for you to do.

00:09:39.730 --> 00:09:42.740
So this has that
same property.

00:09:42.740 --> 00:09:44.930
This is a squared
minus b squared.

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This is a perfect square.

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So then this up here can be
factored as minus 8 times--

00:09:51.720 --> 00:09:55.350
this is a squared
minus b squared.

00:09:55.350 --> 00:10:01.860
So if we could say a
is z and b is 2.

00:10:01.860 --> 00:10:03.940
Because then when we would
have z squared minus 2

00:10:03.940 --> 00:10:05.520
squared, which is 4.

00:10:05.520 --> 00:10:15.180
So if a is z, so it's z plus
2 times z minus 2.

00:10:15.180 --> 00:10:17.250
This is probably the most common
thing you'll see in a

00:10:17.250 --> 00:10:18.720
lot of factoring examples.

00:10:18.720 --> 00:10:20.620
You'll see something that has
this pattern, a squared minus

00:10:20.620 --> 00:10:22.410
b squared, and you should
easily be able

00:10:22.410 --> 00:10:23.120
to recognize it.

00:10:23.120 --> 00:10:25.660
You should be able to prove it
to yourself as well that this

00:10:25.660 --> 00:10:28.850
is the same thing as a plus
b times a minus b.

00:10:28.850 --> 00:10:33.250
Or in this case, z plus
2 times z minus 2.

00:10:33.250 --> 00:10:36.630
And that is z minus 2.

00:10:36.630 --> 00:10:39.040
So this is interesting.

00:10:39.040 --> 00:10:41.900
So I don't see exactly
what we wrote.

00:10:41.900 --> 00:10:44.830
I have a minus 8, but we want
a z plus 2 and z minus 2.

00:10:44.830 --> 00:10:46.140
But they don't have that.

00:10:46.140 --> 00:10:50.400
But what they do have is this--
let me rewrite it.

00:10:50.400 --> 00:10:52.750
They don't have any of it, but
maybe they have-- they want

00:10:52.750 --> 00:10:55.830
the minus to be multiplied by
one of these other terms. So

00:10:55.830 --> 00:10:58.460
what we could do is we could
multiply the minus because

00:10:58.460 --> 00:10:59.820
multiplication is associative.

00:10:59.820 --> 00:11:01.270
It doesn't matter what
order you do it in.

00:11:01.270 --> 00:11:08.270
So we could rewrite this as 8
times z plus 2 times minus 1

00:11:08.270 --> 00:11:11.640
times z minus 2.

00:11:11.640 --> 00:11:15.480
And then this becomes
8 times z plus 2.

00:11:15.480 --> 00:11:17.960
This isn't minus 1, this
is times minus 1.

00:11:17.960 --> 00:11:19.840
Times what?

00:11:19.840 --> 00:11:22.820
Minus z plus 2.

00:11:22.820 --> 00:11:25.900
And now I think that
choice is there.

00:11:25.900 --> 00:11:27.350
Right.

00:11:27.350 --> 00:11:31.790
So we can rewrite this as 8
times 2 plus z-- I'm just

00:11:31.790 --> 00:11:34.980
rearranging it-- times
2 minus z.

00:11:34.980 --> 00:11:38.410
And then that is choice B.

00:11:38.410 --> 00:11:40.980
And so really, given the choices
they gave us, it

00:11:40.980 --> 00:11:43.820
probably would have been faster
to just factor out an 8

00:11:43.820 --> 00:11:45.600
from the beginning and
not a positive 8.

00:11:45.600 --> 00:11:47.800
And you would have immediately
gotten to 2 plus z

00:11:47.800 --> 00:11:49.140
times 2 minus z.

00:11:49.140 --> 00:11:52.010
Anyway, I'll see you
in the next video.


CA Algebra I: Quadratic equation

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.220
We're on problem 53.

00:00:02.220 --> 00:00:04.820
It says Toni is solving this
equation by completing the

00:00:04.820 --> 00:00:08.430
square. ax squared plus bx plus
c is equal to 0, where a

00:00:08.430 --> 00:00:09.325
is greater than 0.

00:00:09.325 --> 00:00:11.525
So this is just a traditional
quadratic right here.

00:00:11.525 --> 00:00:13.550
And let's see what they did.

00:00:13.550 --> 00:00:17.660
First, he subtracted c from
both sides and he got ax

00:00:17.660 --> 00:00:20.950
squared plus bx is
equal to minus c.

00:00:20.950 --> 00:00:22.570
OK, that's fair enough.

00:00:22.570 --> 00:00:23.380
And then let's see.

00:00:23.380 --> 00:00:26.410
He divided both sides by a.

00:00:26.410 --> 00:00:27.530
Right, that's fair enough.

00:00:27.530 --> 00:00:28.800
He got minus c/a.

00:00:28.800 --> 00:00:30.970
Which step should be Step
3 in the solution?

00:00:30.970 --> 00:00:32.159
So he's completing the square.

00:00:32.159 --> 00:00:37.260
So essentially, he wants this
to become a perfect square.

00:00:37.260 --> 00:00:39.870
So let's see how
we can do that.

00:00:39.870 --> 00:00:46.783
So we have x squared plus b/a
x-- and I'm going to leave a

00:00:46.783 --> 00:00:52.020
little space here-- is
equal to minus c/a.

00:00:52.020 --> 00:00:54.110
So for this to be a perfect
square we have to add

00:00:54.110 --> 00:00:56.810
something here, we have
to add a number.

00:00:56.810 --> 00:01:00.190
And we learned from several
videos in the past and we kind

00:01:00.190 --> 00:01:00.860
of pseudo-proved it.

00:01:00.860 --> 00:01:03.710
And actually, I have several
videos I do solely on

00:01:03.710 --> 00:01:04.730
completing the square.

00:01:04.730 --> 00:01:08.010
You essentially have to add
whatever number this is, add

00:01:08.010 --> 00:01:09.830
half of it squared.

00:01:09.830 --> 00:01:12.340
And if that doesn't make sense
to you, watch the Khan Academy

00:01:12.340 --> 00:01:13.680
video on completing
the square.

00:01:13.680 --> 00:01:16.340
But what is half of b/a?

00:01:16.340 --> 00:01:17.590
Well it's b over 2a.

00:01:19.780 --> 00:01:26.510
So 1/2 times b/a is equal
to b over 2a.

00:01:26.510 --> 00:01:28.115
And then, we want to
add this squared.

00:01:28.115 --> 00:01:30.330
So let's add that to both
sides of this equation.

00:01:30.330 --> 00:01:36.860
So we're left with x
squared plus b/a x.

00:01:36.860 --> 00:01:38.830
And we want to add
this squared.

00:01:38.830 --> 00:01:47.980
Plus b over 2a squared is
equal to minus c/a.

00:01:47.980 --> 00:01:49.940
Anything you add to one side of
the equation, you have to

00:01:49.940 --> 00:01:50.580
add to the other.

00:01:50.580 --> 00:01:52.090
So we have to add that
to both sides.

00:01:52.090 --> 00:01:57.220
Plus b over 2a squared.

00:01:57.220 --> 00:01:59.580
And let's see if we've
solved the problem so

00:01:59.580 --> 00:02:00.943
far, what they want.

00:02:00.943 --> 00:02:02.760
X, b over 2-- right.

00:02:02.760 --> 00:02:05.640
This is exactly what we did. x
squared plus b/a plus b over

00:02:05.640 --> 00:02:07.720
2a squared, and they add it to
both sides of the equation.

00:02:07.720 --> 00:02:09.560
So D is the right answer.

00:02:09.560 --> 00:02:11.290
Now if you find that a little
confusing or if it wasn't

00:02:11.290 --> 00:02:12.400
intuitive for you, I
don't want you to

00:02:12.400 --> 00:02:13.350
memorize the steps.

00:02:13.350 --> 00:02:17.730
Watch the Khan Academy video
on completing the square.

00:02:17.730 --> 00:02:19.970
Next problem, 56.

00:02:19.970 --> 00:02:22.480
No, 54.

00:02:22.480 --> 00:02:24.790
All right, this is another one
that should be cut and pasted.

00:02:29.810 --> 00:02:32.490
All right, four steps to derive
the quadratic formula

00:02:32.490 --> 00:02:33.000
are shown below.

00:02:33.000 --> 00:02:36.490
I said in previous videos that
you can derive the quadratic

00:02:36.490 --> 00:02:37.830
formula by completing
the square.

00:02:37.830 --> 00:02:39.090
And we actually do that
in another video.

00:02:39.090 --> 00:02:41.080
I don't want to give too much
of a plug for other videos,

00:02:41.080 --> 00:02:42.410
but let's see what
they want to do.

00:02:42.410 --> 00:02:44.630
What is the correct order
of these steps?

00:02:44.630 --> 00:02:47.500
So the first thing you want to
start off with is just a

00:02:47.500 --> 00:02:48.900
quadratic equation.

00:02:48.900 --> 00:02:52.900
And this one is the
first step.

00:02:52.900 --> 00:02:57.060
This is where we started off
with in the last problem.

00:02:57.060 --> 00:02:59.870
Then what you want to do is
add 1/2 of this squared to

00:02:59.870 --> 00:03:01.210
both sides.

00:03:01.210 --> 00:03:05.040
So b over 2a squared you want
to add to both sides, and

00:03:05.040 --> 00:03:06.360
that's what they did here.

00:03:06.360 --> 00:03:08.012
So our order is I.

00:03:08.012 --> 00:03:10.440
And then you want to do IV.

00:03:10.440 --> 00:03:13.620
That's what we did in
the last problem.

00:03:13.620 --> 00:03:15.860
We did IV.

00:03:15.860 --> 00:03:19.110
And then from here, you know
that this expression right

00:03:19.110 --> 00:03:23.740
here is going to be equal to
x plus b over 2a squared.

00:03:23.740 --> 00:03:25.490
And once again, watch soon.
the completing the squared

00:03:25.490 --> 00:03:26.640
video if that didn't
make sense.

00:03:26.640 --> 00:03:29.170
But the whole reason why you
added this here is so that you

00:03:29.170 --> 00:03:31.780
know that, OK, what two numbers,
when I multiply them

00:03:31.780 --> 00:03:35.480
equal b over 2a squared, and
when I add them equal b/a?

00:03:35.480 --> 00:03:37.250
Well that's obviously,
b over 2a.

00:03:37.250 --> 00:03:39.100
If you add it twice you're
going to get b over a.

00:03:39.100 --> 00:03:40.982
If you square it, you're going
to get this whole expression.

00:03:40.982 --> 00:03:44.560
So you say, oh, this is just x
plus b over 2a squared and you

00:03:44.560 --> 00:03:45.690
get that there.

00:03:45.690 --> 00:03:48.840
And then, is equal to--
and then they just

00:03:48.840 --> 00:03:49.950
simplify this fraction.

00:03:49.950 --> 00:03:52.160
They found a common denominator
and all the rest.

00:03:52.160 --> 00:03:54.170
And so the next step
is Step II.

00:03:54.170 --> 00:03:55.800
And then all you have
left is Step III.

00:03:55.800 --> 00:03:58.970
And you've pretty much derived
the quadratic equation.

00:03:58.970 --> 00:04:00.450
So I, IV, II, III.

00:04:03.410 --> 00:04:04.660
That's choice A.

00:04:06.990 --> 00:04:10.520
Problem 55.

00:04:10.520 --> 00:04:14.450
Which of the solutions--
OK, I'll put all

00:04:14.450 --> 00:04:15.700
of the choices down.

00:04:19.180 --> 00:04:21.089
So which is one of the solutions
to the equation?

00:04:21.089 --> 00:04:22.670
So immediately when you see all
of the choices, they have

00:04:22.670 --> 00:04:24.170
these square roots
and all that.

00:04:24.170 --> 00:04:25.420
This isn't something that
you would factor.

00:04:25.420 --> 00:04:26.950
You would use a quadratic
equation here.

00:04:26.950 --> 00:04:27.670
So let's do that.

00:04:27.670 --> 00:04:34.510
So the quadratic equation is, so
if this is Ax squared plus

00:04:34.510 --> 00:04:37.370
Bx plus C is equal to 0.

00:04:37.370 --> 00:04:40.270
The quadratic equation
is minus b.

00:04:40.270 --> 00:04:41.270
Well they do it lowercase.

00:04:41.270 --> 00:04:47.280
Plus or minus the square root of
b squared minus 4ac, all of

00:04:47.280 --> 00:04:48.500
that over 2a.

00:04:48.500 --> 00:04:51.250
And this is just derived from
completing the square with

00:04:51.250 --> 00:04:53.440
this, but we do that
in another video.

00:04:53.440 --> 00:04:54.540
And so let's substitute it in.

00:04:54.540 --> 00:04:56.170
What is b?

00:04:56.170 --> 00:04:58.310
b is minus 1, right?

00:04:58.310 --> 00:05:01.870
So minus minus 1, that's
a positive 1.

00:05:01.870 --> 00:05:05.220
Plus or minus the square
root of b squared.

00:05:05.220 --> 00:05:08.290
Minus 1 squared is 1.

00:05:08.290 --> 00:05:12.220
Minus 4 times a.

00:05:12.220 --> 00:05:13.560
a is 2.

00:05:13.560 --> 00:05:15.030
Times 2.

00:05:15.030 --> 00:05:16.040
Times c.

00:05:16.040 --> 00:05:18.080
c is minus 4.

00:05:18.080 --> 00:05:21.690
So times minus 4.

00:05:21.690 --> 00:05:23.930
All of that over 2a.

00:05:23.930 --> 00:05:25.995
a is 2, so 2 times a is 4.

00:05:25.995 --> 00:05:31.540
So that becomes 1 plus or
minus the square root.

00:05:31.540 --> 00:05:32.560
So we have a 1.

00:05:32.560 --> 00:05:36.170
So we have minus 4 times
a 2 times a minus 4.

00:05:36.170 --> 00:05:39.530
That's the same thing as a plus
4 times 2 times a plus 4.

00:05:39.530 --> 00:05:41.490
Let's just take that
minus out.

00:05:41.490 --> 00:05:42.490
So it's plus.

00:05:42.490 --> 00:05:45.210
There's no minus here.

00:05:45.210 --> 00:05:47.670
So let's see, 4 times 2 is 8.

00:05:47.670 --> 00:05:48.860
Times 4 is 32.

00:05:48.860 --> 00:05:52.140
Plus 1 is 33.

00:05:52.140 --> 00:05:53.840
All of that over 4.

00:05:53.840 --> 00:05:56.030
Let's see, we're not
quite there yet.

00:05:56.030 --> 00:05:58.740
Well they say, which is one of
the solutions to the equation?

00:05:58.740 --> 00:06:00.200
So let's see.

00:06:00.200 --> 00:06:03.080
If we wanted to simplify
this out a-- well,

00:06:03.080 --> 00:06:04.770
this is right here.

00:06:04.770 --> 00:06:06.800
Because we have 1 plus
or minus the square

00:06:06.800 --> 00:06:07.750
root of 33 over 4.

00:06:07.750 --> 00:06:08.740
Well they wrote just
one of them.

00:06:08.740 --> 00:06:11.300
They wrote just the plus.

00:06:11.300 --> 00:06:12.800
So C is one of the solutions.

00:06:12.800 --> 00:06:15.310
The other one would have been if
you had a minus sign here.

00:06:15.310 --> 00:06:17.600
Anyway, next problem.

00:06:17.600 --> 00:06:24.810
56.

00:06:24.810 --> 00:06:26.960
And this is another one I
need to cut and paste.

00:06:29.670 --> 00:06:33.280
It says, which statement best
explains why there's no real

00:06:33.280 --> 00:06:36.010
solution to the quadratic
equation?

00:06:36.010 --> 00:06:39.670
OK, so I already have
a guess of why this

00:06:39.670 --> 00:06:40.960
won't have a solution.

00:06:40.960 --> 00:06:43.700
But in general-- well, let's
try the quadratic equation.

00:06:43.700 --> 00:06:44.870
Before even looking
at this problem,

00:06:44.870 --> 00:06:45.480
let's get an intuition.

00:06:45.480 --> 00:06:49.260
It's negative b plus or minus
you the square root of b

00:06:49.260 --> 00:06:55.520
squared minus 4ac, all
of that over 2a.

00:06:55.520 --> 00:06:59.310
My question is to you, when does
this not make any sense?

00:06:59.310 --> 00:07:02.075
Well you know, this'll work
for any b, any 2a.

00:07:02.075 --> 00:07:05.400
But when does the square root
sign really fall apart, at

00:07:05.400 --> 00:07:06.920
least when we're dealing
with real numbers,

00:07:06.920 --> 00:07:08.200
and that's a clue?

00:07:08.200 --> 00:07:12.050
Well, it's when you have a
negative number under here.

00:07:12.050 --> 00:07:13.820
If you end up with a negative
number under the square root

00:07:13.820 --> 00:07:16.230
sign, at least if we haven't
learned imaginary numbers yet,

00:07:16.230 --> 00:07:17.810
you don't know what to do.

00:07:17.810 --> 00:07:20.320
There's no real solution to
the quadratic equation.

00:07:20.320 --> 00:07:25.210
So if b squared minus
4ac is less than

00:07:25.210 --> 00:07:26.810
0, you're in trouble.

00:07:26.810 --> 00:07:28.540
There's no real solution.

00:07:28.540 --> 00:07:30.420
You can't take a square root of
a negative sign if you're

00:07:30.420 --> 00:07:32.090
doing with real numbers.

00:07:32.090 --> 00:07:34.730
So that's probably going
to be the problem here.

00:07:34.730 --> 00:07:36.970
So let's see what b squared
minus 4ac is.

00:07:36.970 --> 00:07:38.410
You have b is 1.

00:07:38.410 --> 00:07:44.320
So 1 minus 4 times a.

00:07:44.320 --> 00:07:46.410
a is 2.

00:07:46.410 --> 00:07:49.030
2 times c is 7.

00:07:49.030 --> 00:07:51.680
And sure enough, 1 times 4 times
2 times 7 is going to be

00:07:51.680 --> 00:07:53.330
less than 0.

00:07:53.330 --> 00:07:55.610
So let's just see what
they have here.

00:07:55.610 --> 00:07:57.890
Right, the value of 1
squared-- oh, right.

00:07:57.890 --> 00:07:59.080
It's b squared.

00:07:59.080 --> 00:08:00.520
Well 1 squared, same
thing as 1.

00:08:00.520 --> 00:08:03.000
1 squared minus 4
times 2 times 7,

00:08:03.000 --> 00:08:04.090
sure enough is negative.

00:08:04.090 --> 00:08:06.150
So that's why we don't
have a real

00:08:06.150 --> 00:08:09.270
solution to this equation.

00:08:09.270 --> 00:08:10.100
Next problem.

00:08:10.100 --> 00:08:11.350
I'm actually out of space.

00:08:15.520 --> 00:08:17.360
OK, they want to know
the solution set to

00:08:17.360 --> 00:08:18.400
this quadratic equation.

00:08:18.400 --> 00:08:20.210
I'll just copy and paste.

00:08:22.960 --> 00:08:25.250
So that's essentially the
set of the x's that

00:08:25.250 --> 00:08:28.020
satisfy this equation.

00:08:28.020 --> 00:08:30.480
And obviously, for any x that
you put in this, the left-hand

00:08:30.480 --> 00:08:31.630
side is going to
be equal to 0.

00:08:31.630 --> 00:08:32.990
So what x's are valid?

00:08:32.990 --> 00:08:34.929
And they just want us to apply
the quadratic equation.

00:08:34.929 --> 00:08:37.600
So we've written it a couple of
times, but let's just do it

00:08:37.600 --> 00:08:38.250
straight up.

00:08:38.250 --> 00:08:39.919
So it's negative b.

00:08:39.919 --> 00:08:41.169
b is 2.

00:08:41.169 --> 00:08:44.169
So it's negative 2
plus or minus the

00:08:44.169 --> 00:08:45.740
square root of b squared.

00:08:45.740 --> 00:08:47.970
Well that's 2 squared.

00:08:47.970 --> 00:08:51.520
Minus 4 times a.

00:08:51.520 --> 00:08:53.410
a is 8.

00:08:53.410 --> 00:08:56.070
Times c, which is 1.

00:08:56.070 --> 00:08:58.560
All of that over 2 times a.

00:08:58.560 --> 00:09:03.800
So 2 times 8, which is equal to
minus 2 plus or minus the

00:09:03.800 --> 00:09:11.215
square root of 4-- let's see.

00:09:11.215 --> 00:09:13.040
Did I write this down?

00:09:13.040 --> 00:09:21.160
Negative b plus or minus the
square root of b squared minus

00:09:21.160 --> 00:09:23.650
4 times a times c.

00:09:23.650 --> 00:09:24.270
Right.

00:09:24.270 --> 00:09:28.890
So you get 4 minus 32.

00:09:28.890 --> 00:09:30.900
That's why I was double checking
to see if I did this

00:09:30.900 --> 00:09:32.460
right because I'm going to get
a negative number here.

00:09:32.460 --> 00:09:34.700
All of that over 16.

00:09:34.700 --> 00:09:36.690
And so we're going to end up
with the same conundrum we had

00:09:36.690 --> 00:09:39.480
in the last. 4 minus 32, we're
going to end with minus 2 plus

00:09:39.480 --> 00:09:43.785
or minus the square root
of minus 28 over 16.

00:09:43.785 --> 00:09:46.200
And if we're dealing with real
numbers, I mean there's no

00:09:46.200 --> 00:09:47.080
real solution here.

00:09:47.080 --> 00:09:47.950
And at first I was worried.

00:09:47.950 --> 00:09:49.650
I thought I made a careless
mistake or there was an error

00:09:49.650 --> 00:09:50.360
in the problem.

00:09:50.360 --> 00:09:52.460
But then I look at
the choices.

00:09:52.460 --> 00:09:53.440
They have choice D.

00:09:53.440 --> 00:09:56.500
And I'll copy and paste
choice D here.

00:09:56.500 --> 00:09:57.350
Choice D.

00:09:57.350 --> 00:09:58.370
No real solution.

00:09:58.370 --> 00:10:00.780
So that's the answer, because
you can't take a square root

00:10:00.780 --> 00:10:05.860
of a negative number and stay
in the set of real numbers.

00:10:05.860 --> 00:10:07.620
Let's see, do I have time
for another one?

00:10:07.620 --> 00:10:09.910
I'm over the 10 minutes.

00:10:09.910 --> 00:10:11.330
I'll wait for the next video.

00:10:11.330 --> 00:10:12.580
See


CA Algebra I: Completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:02.280
We're on problem 48.

00:00:02.280 --> 00:00:05.800
It says if x squared is added
to x, the sum is 42.

00:00:05.800 --> 00:00:07.220
So let's just write that out.

00:00:07.220 --> 00:00:13.390
If x squared is added to x,
the sum is equal to 42.

00:00:13.390 --> 00:00:16.079
Which of the following could
be the value of x?

00:00:16.079 --> 00:00:17.660
So essentially they just want
us to solve this equation.

00:00:17.660 --> 00:00:22.930
So the easiest way to do it is
to write it as a quadratic

00:00:22.930 --> 00:00:24.500
equaling 0 and then
factoring it.

00:00:24.500 --> 00:00:30.130
So we could write this as x
squared plus x minus 42 is

00:00:30.130 --> 00:00:31.560
equal to 0.

00:00:31.560 --> 00:00:32.020
And let's think.

00:00:32.020 --> 00:00:36.040
What two numbers when I add
them equal 1, and when I

00:00:36.040 --> 00:00:38.710
multiply them equal minus 42?

00:00:38.710 --> 00:00:42.110
And the fact that when I
multiply them equals minus 42

00:00:42.110 --> 00:00:44.530
tells me that one of them has to
be positive and one of them

00:00:44.530 --> 00:00:45.290
has to be negative.

00:00:45.290 --> 00:00:47.280
That's the only way that when
you multiply two numbers

00:00:47.280 --> 00:00:48.700
you're going to have
a negative number.

00:00:48.700 --> 00:00:50.500
So one of them has to be
positive, one of them has to

00:00:50.500 --> 00:00:51.370
be negative.

00:00:51.370 --> 00:00:53.640
And so when we're adding a
positive and a negative,

00:00:53.640 --> 00:00:55.430
you're really finding the
difference between the two.

00:00:55.430 --> 00:00:58.080
So the difference between the
two numbers has to be 1 and

00:00:58.080 --> 00:01:00.000
their product has to be 42.

00:01:00.000 --> 00:01:01.610
And I noticed when I see
42 I immediately

00:01:01.610 --> 00:01:02.770
think, oh, 6 and 7.

00:01:02.770 --> 00:01:04.930
6 times 7 is 42.

00:01:04.930 --> 00:01:07.850
And since when you add them
you get a positive 1, 7's

00:01:07.850 --> 00:01:11.780
probably the positive one and
minus 6 or 6 is probably the

00:01:11.780 --> 00:01:12.730
negative one.

00:01:12.730 --> 00:01:14.210
So let's try it out.

00:01:14.210 --> 00:01:21.190
x plus 7 times x minus
6 equal to 0.

00:01:21.190 --> 00:01:26.460
And right, 7 times minus
6 is minus 42.

00:01:26.460 --> 00:01:30.460
It's really 7x plus minus 6x
is equal to positive x.

00:01:30.460 --> 00:01:33.740
Or you could think 7 plus
minus 6 is equal to the

00:01:33.740 --> 00:01:36.466
coefficient on the
x, which is 1.

00:01:36.466 --> 00:01:37.890
But anyway, that works.

00:01:37.890 --> 00:01:40.050
And you could multiply this
out and try it out.

00:01:40.050 --> 00:01:42.860
And everything I'm saying,
it's not some voodoo.

00:01:42.860 --> 00:01:46.200
The reason why I say that they
have to add up to the 1 is

00:01:46.200 --> 00:01:48.620
because when you multiply
this out, that's what

00:01:48.620 --> 00:01:49.870
builds up this term.

00:01:49.870 --> 00:01:54.370
This 7 times x plus the minus
6 times the other x.

00:01:54.370 --> 00:01:56.880
That's what builds this term
when you multiply it out.

00:01:56.880 --> 00:01:59.010
This term comes from
the x times the x.

00:01:59.010 --> 00:02:02.620
The minus 42 comes from the
7 times the minus 6.

00:02:02.620 --> 00:02:04.640
Anyway, now we're
at this point.

00:02:04.640 --> 00:02:07.210
We say OK, well how do we get--
we have two things when

00:02:07.210 --> 00:02:08.840
you multiply them equal to 0.

00:02:08.840 --> 00:02:11.120
Well that means that one of them
or both of them have to

00:02:11.120 --> 00:02:12.110
be equal to 0.

00:02:12.110 --> 00:02:16.520
So that means that x plus 7 is
equal to 0, which would mean

00:02:16.520 --> 00:02:17.890
subtract 7 from both sides.

00:02:17.890 --> 00:02:20.290
Which means x is equal
to minus 7.

00:02:20.290 --> 00:02:23.580
Or x minus 6 is equal to 0.

00:02:23.580 --> 00:02:26.090
Add 6 to both sides. x is
equal to 6. x would

00:02:26.090 --> 00:02:28.110
be 6 or minus 7.

00:02:28.110 --> 00:02:29.830
And they have one of
the choices there,

00:02:29.830 --> 00:02:32.710
which was choice A.

00:02:32.710 --> 00:02:34.340
Next problem.

00:02:34.340 --> 00:02:35.820
49.

00:02:35.820 --> 00:02:38.740
What quantity should be added to
both sides of this equation

00:02:38.740 --> 00:02:40.320
to complete the square?

00:02:40.320 --> 00:02:41.930
So when you complete the square
you want the thing to

00:02:41.930 --> 00:02:44.740
just look like a-- you want
whatever's on the left-hand

00:02:44.740 --> 00:02:46.190
side to look like a
perfect square.

00:02:46.190 --> 00:02:47.350
And what do I mean by
perfect square?

00:02:47.350 --> 00:02:54.580
So if I had x plus a squared,
that's equal to x plus a

00:02:54.580 --> 00:02:57.030
times x plus a.

00:02:57.030 --> 00:03:00.370
And that's equal to x
times x, x squared.

00:03:00.370 --> 00:03:03.890
x times that a, so
that's plus ax.

00:03:03.890 --> 00:03:05.680
And now this a times this x.

00:03:05.680 --> 00:03:07.250
So that's another ax.

00:03:07.250 --> 00:03:08.730
Plus this a times that a.

00:03:08.730 --> 00:03:10.410
So plus a squared.

00:03:10.410 --> 00:03:12.100
And that's equal to x squared.

00:03:12.100 --> 00:03:17.720
Plus-- we have two of these
now-- plus 2ax plus a squared.

00:03:17.720 --> 00:03:21.520
So essentially we want this, we
want the left-hand side to

00:03:21.520 --> 00:03:22.930
have this form.

00:03:22.930 --> 00:03:24.330
So we say, this is
a perfect square.

00:03:24.330 --> 00:03:26.742
We can say that's the same thing
as x plus a squared.

00:03:26.742 --> 00:03:29.820
So let's think about
how we can do it.

00:03:29.820 --> 00:03:37.040
If we have x squared minus 8x
is equal to 5 and I put a

00:03:37.040 --> 00:03:39.640
space here for a reason because
we want to add or

00:03:39.640 --> 00:03:42.490
subtract something here so it
looks like a perfect square.

00:03:42.490 --> 00:03:44.530
So think about it.

00:03:44.530 --> 00:03:48.600
When we have this format, in
order for this thing to be a

00:03:48.600 --> 00:03:53.820
perfect square, whatever this
coefficient is right here,

00:03:53.820 --> 00:03:57.430
this term right here has to
be half of this, squared.

00:03:57.430 --> 00:04:00.310
a squared is half
of 2a squared.

00:04:00.310 --> 00:04:03.990
So if we took half of minus
8, that's minus 4.

00:04:03.990 --> 00:04:06.890
In this case, if we said
2a is equal to 8, a

00:04:06.890 --> 00:04:08.720
would be minus 4.

00:04:08.720 --> 00:04:10.910
And so minus 4 squared
is what?

00:04:10.910 --> 00:04:14.830
It's plus 16.

00:04:14.830 --> 00:04:15.990
And this is an equation.

00:04:15.990 --> 00:04:18.005
So one you do to one side of an
equation you have to do to

00:04:18.005 --> 00:04:18.955
the other side of
the equation.

00:04:18.955 --> 00:04:21.570
So you have to say that
that is also equal to.

00:04:21.570 --> 00:04:22.820
So you have to add
16 to both sides.

00:04:22.820 --> 00:04:26.600
Otherwise you're changing
the equation.

00:04:26.600 --> 00:04:28.610
Now this, hopefully you
recognize this as already a

00:04:28.610 --> 00:04:29.300
perfect square.

00:04:29.300 --> 00:04:31.270
I mean you could look at this
pattern up here or you could

00:04:31.270 --> 00:04:35.090
say, OK, if I add minus 4 to
itself twice I get minus 8.

00:04:35.090 --> 00:04:37.680
If I multiply it by
itself I get 16.

00:04:37.680 --> 00:04:41.790
So this is x minus 4 squared.

00:04:41.790 --> 00:04:44.490
That's equal to 25.

00:04:44.490 --> 00:04:46.370
And actually, just if you're
curious-- and we did this in

00:04:46.370 --> 00:04:48.170
the Khan Academy, we did a
couple of videos on this--

00:04:48.170 --> 00:04:50.400
this is how you prove the
quadratic equation.

00:04:50.400 --> 00:04:54.050
You essentially complete the
square with arbitrary numbers

00:04:54.050 --> 00:04:56.320
a, b, and c, and you get
the quadratic equation.

00:04:56.320 --> 00:04:59.540
You know, we show it in 10
minutes, so it's not this

00:04:59.540 --> 00:05:02.270
impossibly hard thing
to understand.

00:05:02.270 --> 00:05:04.860
They just want to know, what
do you add to both sides of

00:05:04.860 --> 00:05:06.220
this equation?

00:05:06.220 --> 00:05:08.500
What quantity should be added to
both sides of this equation

00:05:08.500 --> 00:05:09.370
to complete the square?

00:05:09.370 --> 00:05:12.100
So the answer to this
one was 16.

00:05:12.100 --> 00:05:14.670
But they just as equally could
have said, solve it by

00:05:14.670 --> 00:05:15.860
completing the square.

00:05:15.860 --> 00:05:18.730
And you'd say oh, x minus 4
squared is equal to 25.

00:05:18.730 --> 00:05:22.160
So x minus 4 is equal
to plus or minus 5.

00:05:22.160 --> 00:05:24.310
And then you could say,
x is equal to plus or

00:05:24.310 --> 00:05:26.690
minus 5 plus 4.

00:05:26.690 --> 00:05:29.930
And then you could say,
OK, which is 4 plus

00:05:29.930 --> 00:05:31.190
positive 5 is 9.

00:05:31.190 --> 00:05:35.780
4 plus minus 5 is--
or, minus 1.

00:05:35.780 --> 00:05:37.160
Anyway, they didn't ask us
that, so we don't have to

00:05:37.160 --> 00:05:39.350
spend too much time
thinking about it.

00:05:42.430 --> 00:05:49.310
Let's see, we're
on problem 50.

00:05:49.310 --> 00:05:53.070
Let me see, problem 50.

00:05:53.070 --> 00:06:01.970
I'll copy and paste 50 and 51.

00:06:01.970 --> 00:06:05.190
All right, what are the
solutions for the quadratic

00:06:05.190 --> 00:06:12.190
equation x squared plus
6x is equal to 16?

00:06:12.190 --> 00:06:15.350
And the temptation here is
really to kind of try to solve

00:06:15.350 --> 00:06:16.870
it the way you do a
linear equation.

00:06:16.870 --> 00:06:19.110
I don't know, factor out an
x and-- I don't know, do

00:06:19.110 --> 00:06:19.580
whatever else.

00:06:19.580 --> 00:06:21.010
But the important thing to
recognize is this is a

00:06:21.010 --> 00:06:22.220
quadratic equation.

00:06:22.220 --> 00:06:24.910
And the easiest way to solve it
is to put all the terms on

00:06:24.910 --> 00:06:26.660
one side and then get a
0 on the other side.

00:06:26.660 --> 00:06:28.640
And then either factor
it or use the

00:06:28.640 --> 00:06:30.090
actual quadratic equation.

00:06:30.090 --> 00:06:32.300
Or complete the square, whatever
you need to do.

00:06:32.300 --> 00:06:34.970
So let's subtract 16
from both sides.

00:06:34.970 --> 00:06:40.230
And you get x squared plus 6x
minus 16 is equal to 0.

00:06:40.230 --> 00:06:43.270
I just subtracted 16 from
both sides to get here.

00:06:43.270 --> 00:06:45.600
And before just jumping into the
quadratic equation, let's

00:06:45.600 --> 00:06:47.270
see if we can factor
it by inspection.

00:06:47.270 --> 00:06:54.165
So what two numbers, when I
add them, equal 6-- and we

00:06:54.165 --> 00:06:58.602
want positive 6-- and when I
multiply them equal minus 16?

00:06:58.602 --> 00:07:01.360
And once again, since it's a
minus 16, if you multiply two

00:07:01.360 --> 00:07:03.310
numbers you get a
negative number.

00:07:03.310 --> 00:07:04.530
They have to be different
signs.

00:07:04.530 --> 00:07:07.560
One has to be positive and
one has to be negative.

00:07:07.560 --> 00:07:10.780
And their difference will be 6
because one's positive and

00:07:10.780 --> 00:07:11.435
one's negative.

00:07:11.435 --> 00:07:12.990
So let me think about it.

00:07:12.990 --> 00:07:16.470
So if I had minus-- well,
8 and 2 is equal to 16.

00:07:16.470 --> 00:07:17.390
And they're 6 apart.

00:07:17.390 --> 00:07:19.800
So if I did plus 8 and
minus 2-- right.

00:07:19.800 --> 00:07:22.370
Plus 8 and minus 2
is positive 6.

00:07:22.370 --> 00:07:27.360
So it's x plus 8 times
x minus 2.

00:07:27.360 --> 00:07:28.620
And that really just takes
a lot of practice.

00:07:28.620 --> 00:07:29.870
You say, OK, what two numbers?

00:07:29.870 --> 00:07:30.880
16.

00:07:30.880 --> 00:07:31.340
OK.

00:07:31.340 --> 00:07:32.010
8 and 2.

00:07:32.010 --> 00:07:34.290
Well they're going to have
to be different signs.

00:07:34.290 --> 00:07:36.480
But I have a positive one here,
so whichever number's

00:07:36.480 --> 00:07:38.430
larger is probably going
to be the positive one.

00:07:38.430 --> 00:07:40.120
So positive 8 and minus 2.

00:07:40.120 --> 00:07:41.790
Yeah, when you add them up,
they equal minus 6.

00:07:41.790 --> 00:07:43.660
Yeah, it works.

00:07:43.660 --> 00:07:46.120
So you set that equal to 0.

00:07:46.120 --> 00:07:48.910
And you say OK, this has to be
equal to 0, or that has to be

00:07:48.910 --> 00:07:51.320
equal to 0.

00:07:51.320 --> 00:07:54.220
So x is either equal
to minus 8.

00:07:54.220 --> 00:07:57.830
If you say x plus 8 is equal to
0 then subtract 8 from both

00:07:57.830 --> 00:07:59.590
sides, you get x is
equal to minus 8.

00:07:59.590 --> 00:08:01.075
I shouldn't have skipped
that step, but

00:08:01.075 --> 00:08:02.030
I'll do the step here.

00:08:02.030 --> 00:08:04.530
Or you could say x minus
2 is equal to 0.

00:08:04.530 --> 00:08:07.220
Add 2 to both sides, you
get x is equal to 2.

00:08:07.220 --> 00:08:10.070
What x makes this
term equal to 0?

00:08:10.070 --> 00:08:11.330
And you could look at
it from inspection.

00:08:11.330 --> 00:08:19.190
So x could either be minus 8
or 2, and that is choice C.

00:08:19.190 --> 00:08:21.290
Problem 51.

00:08:21.290 --> 00:08:24.970
Leanne correctly solved the
equation x squared plus 4x

00:08:24.970 --> 00:08:27.020
equals 6 by completing
the square.

00:08:27.020 --> 00:08:29.530
Which equation is part
of her solution?

00:08:29.530 --> 00:08:31.385
OK, so the same thing.

00:08:31.385 --> 00:08:34.010
x squared plus 4x.

00:08:34.010 --> 00:08:35.260
And when you complete the square
you're going to add

00:08:35.260 --> 00:08:35.809
something here.

00:08:35.809 --> 00:08:36.954
So I'm going to leave
a little blank.

00:08:36.954 --> 00:08:38.850
Is equal to 6.

00:08:38.850 --> 00:08:41.880
So what could I add here that
makes this expression look

00:08:41.880 --> 00:08:43.950
like a perfect square?

00:08:43.950 --> 00:08:45.320
Well, got to the pattern
that we did a

00:08:45.320 --> 00:08:46.850
couple of problems ago.

00:08:46.850 --> 00:08:50.840
Whatever's here should be the
square of half of this.

00:08:50.840 --> 00:08:53.510
So 4-- well, half
of that is 2.

00:08:53.510 --> 00:08:54.570
2 squared is 4.

00:08:54.570 --> 00:08:56.400
So I should add 4
to that side.

00:08:56.400 --> 00:08:57.630
If I add 4 to that side,
I have to add 4 to

00:08:57.630 --> 00:08:59.490
this side as well.

00:08:59.490 --> 00:09:02.080
And now this 2 plus
2 is equal to 4.

00:09:02.080 --> 00:09:03.800
2 times 2 is equal to 4.

00:09:03.800 --> 00:09:06.340
So this is x plus 2 squared.

00:09:06.340 --> 00:09:07.660
And I really want you to
get the intuition.

00:09:07.660 --> 00:09:09.750
Don't memorize the steps for
completing the square.

00:09:09.750 --> 00:09:12.380
I want you to really
understand why.

00:09:12.380 --> 00:09:14.120
This is the square
of half of that.

00:09:14.120 --> 00:09:15.230
And we showed it in
the beginning.

00:09:15.230 --> 00:09:18.650
Square a lot of binomials and
see for yourself that that's

00:09:18.650 --> 00:09:20.270
always going to be the case.

00:09:20.270 --> 00:09:22.200
Anyway, so this is
x plus 2 squared.

00:09:22.200 --> 00:09:25.360
That's going to be equal to--
6 plus 4 is equal to 10.

00:09:25.360 --> 00:09:28.170
And that is choice B.

00:09:30.842 --> 00:09:34.760
I think we have time
for one more.

00:09:34.760 --> 00:09:40.980
One more problem, problem 52.

00:09:40.980 --> 00:09:45.330
Copied it and now I
have pasted it.

00:09:45.330 --> 00:09:47.410
Carter is solving this equation
by factoring.

00:09:47.410 --> 00:09:51.610
Which expression could be one
of his correct factors?

00:09:51.610 --> 00:09:58.470
Once again, I like to personally
separate out the

00:09:58.470 --> 00:10:00.030
number that goes into
all of them.

00:10:00.030 --> 00:10:01.320
And all of these are
divisible by 5.

00:10:01.320 --> 00:10:03.260
And that just simplifies
it in my head.

00:10:03.260 --> 00:10:05.760
So if I divide all of these by
5-- actually, I could just

00:10:05.760 --> 00:10:08.600
divide both sides of
this equation by 5.

00:10:08.600 --> 00:10:10.410
0 divided by 5 is 0.

00:10:10.410 --> 00:10:14.910
And then that left side divided
by 5 becomes 2x

00:10:14.910 --> 00:10:27.960
squared minus 5x plus
3 is equal to 0.

00:10:27.960 --> 00:10:30.400
So if this is 2x squared here,
so it's going to be two

00:10:30.400 --> 00:10:33.880
numbers when you multiply them
equal 3 and when you-- so

00:10:33.880 --> 00:10:36.210
let's think about this
a little bit.

00:10:36.210 --> 00:10:37.830
Actually, let me write it down
here because I think I'll need

00:10:37.830 --> 00:10:38.930
more space.

00:10:38.930 --> 00:10:44.240
2x squared minus 5x plus
3 is equal to 0.

00:10:44.240 --> 00:10:46.040
And I just divided both
sides of the equation

00:10:46.040 --> 00:10:47.755
by 5 to get to this.

00:10:47.755 --> 00:10:50.430
So let's see what
we can do here.

00:10:50.430 --> 00:10:52.870
So we have a 2x squared here
and they already kind of

00:10:52.870 --> 00:10:54.860
hinted to us that we're going
to have an integer solution,

00:10:54.860 --> 00:10:56.230
so we can factor this.

00:10:56.230 --> 00:11:00.460
So the intuition is that this is
going to be 2x times-- you

00:11:00.460 --> 00:11:02.250
know, plus something.

00:11:02.250 --> 00:11:03.460
Plus a.

00:11:03.460 --> 00:11:05.650
Times-- well what times?

00:11:05.650 --> 00:11:06.810
Times probably x, right?

00:11:06.810 --> 00:11:09.340
2x times x is 2x squared.

00:11:09.340 --> 00:11:11.050
Now that wouldn't be completely
obvious if they

00:11:11.050 --> 00:11:12.970
didn't already tell us that
we could factor this.

00:11:12.970 --> 00:11:13.550
You might have to use a

00:11:13.550 --> 00:11:14.840
quadratic equation or something.

00:11:14.840 --> 00:11:16.660
Actually, the quadratic equation
wouldn't be something

00:11:16.660 --> 00:11:18.800
crazy to use here because
you can just

00:11:18.800 --> 00:11:19.890
kind of plug and chug.

00:11:19.890 --> 00:11:21.210
But let's see if we can
get the intuition.

00:11:21.210 --> 00:11:25.470
So it's going to be 2x plus
something times x plus

00:11:25.470 --> 00:11:26.830
something else.

00:11:26.830 --> 00:11:30.310
If we were to multiply this out,
you get 2x times x is 2x

00:11:30.310 --> 00:11:31.380
squared as it should.

00:11:31.380 --> 00:11:35.250
2x times b is plus 2bx.

00:11:35.250 --> 00:11:38.180
a times x is plus ax.

00:11:38.180 --> 00:11:42.320
a times b is plus ab.

00:11:42.320 --> 00:11:43.570
And so let's see what we get.

00:11:47.770 --> 00:11:55.400
So plus 2b plus ax plus ab.

00:11:55.400 --> 00:11:57.420
2x squared.

00:11:57.420 --> 00:11:59.180
OK, now can do pattern
matching.

00:11:59.180 --> 00:12:01.440
This was our original thing.

00:12:01.440 --> 00:12:05.890
So 2 times b plus a has to be
equal to-- this term is the

00:12:05.890 --> 00:12:08.150
same thing as this
term right here.

00:12:08.150 --> 00:12:10.080
And that term has to be
the same thing as

00:12:10.080 --> 00:12:13.440
that term right there.

00:12:13.440 --> 00:12:17.230
So first of all, I have
a positive 3 here.

00:12:17.230 --> 00:12:19.290
So I'm multiplying two numbers
to get a positive 3.

00:12:19.290 --> 00:12:23.940
So they either have to be both
positive or both negative.

00:12:23.940 --> 00:12:26.460
And then the other interesting
thing is we have-- when I take

00:12:26.460 --> 00:12:28.070
2 times one of them plus
the other one, I

00:12:28.070 --> 00:12:29.920
get a negative number.

00:12:29.920 --> 00:12:32.520
So the only way when you're
dealing with negative numbers,

00:12:32.520 --> 00:12:34.750
and when you just multiply it
times a positive and add them

00:12:34.750 --> 00:12:37.050
to each other you get another
negative number, is if they're

00:12:37.050 --> 00:12:38.340
both negative.

00:12:38.340 --> 00:12:39.860
This told us they both
have to be negative

00:12:39.860 --> 00:12:41.050
because this is positive.

00:12:41.050 --> 00:12:44.460
And then since when you add
them without any negative

00:12:44.460 --> 00:12:46.510
signs you get a negative number,
it tells you that that

00:12:46.510 --> 00:12:47.920
has to be negative as well.

00:12:47.920 --> 00:12:49.190
So let's see.

00:12:49.190 --> 00:12:54.390
Let's just try 3-- negative
3 and negative 1.

00:12:54.390 --> 00:12:56.300
If negative 3 and negative 1.

00:12:56.300 --> 00:12:56.800
So you're right.

00:12:56.800 --> 00:12:57.330
Yeah.

00:12:57.330 --> 00:13:05.450
If b is equal to minus 1 and a
is equal to minus 3, then 2

00:13:05.450 --> 00:13:08.050
times minus 1 is minus 2.

00:13:08.050 --> 00:13:09.080
Minus 3.

00:13:09.080 --> 00:13:12.460
Right, so b is equal to minus
1 and a is equal to minus 3.

00:13:12.460 --> 00:13:14.740
This is a bit of an
art form here.

00:13:14.740 --> 00:13:16.950
I mean there isn't like a plug
and chug, very mechanical way

00:13:16.950 --> 00:13:17.680
of doing this.

00:13:17.680 --> 00:13:20.610
The quadratic equation is one,
but this is the best way that,

00:13:20.610 --> 00:13:23.220
at least, I know how to
do these without.

00:13:23.220 --> 00:13:24.980
So, we know what a and b is.

00:13:24.980 --> 00:13:28.460
So it's 2x-- a is minus 3.

00:13:28.460 --> 00:13:34.910
2x minus 3 times x plus
b. b is minus 1.

00:13:34.910 --> 00:13:38.290
So that's the factorization.

00:13:38.290 --> 00:13:41.590
So 2x minus 3 times x
minus 1, which one?

00:13:41.590 --> 00:13:43.180
They have this one right here.

00:13:43.180 --> 00:13:45.090
2x minus 3.

00:13:45.090 --> 00:13:46.070
And I'm all out of time.

00:13:46.070 --> 00:13:47.790
See you in the next video.


CA Algebra I: Systems of Inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:05.980
We're on problem 33 and it
asks, which equation

00:00:05.980 --> 00:00:10.760
represents a line that is
parallel to y is equal to

00:00:10.760 --> 00:00:15.490
minus 5/4 x plus 2?

00:00:15.490 --> 00:00:18.100
So a line that's parallel will
have the same slope.

00:00:18.100 --> 00:00:19.980
And we can just look at this, we
could inspect this and say,

00:00:19.980 --> 00:00:23.780
OK, the slope of this line is
the coefficient on the x term.

00:00:23.780 --> 00:00:26.260
So slope on this line
is right there.

00:00:26.260 --> 00:00:31.580
Because the slope is
equal to minus 5/4.

00:00:31.580 --> 00:00:33.400
And so we just have to look at
the other choices and say,

00:00:33.400 --> 00:00:36.570
which one has the same
slope of minus 5/4?

00:00:36.570 --> 00:00:40.850
And choice A actually is, choice
A is y is equal to

00:00:40.850 --> 00:00:45.880
minus 5/4 x plus 1.

00:00:45.880 --> 00:00:47.720
So it has the exact same
slope and it's just

00:00:47.720 --> 00:00:50.370
shifted down by 1.

00:00:50.370 --> 00:00:51.795
So A is our choice.

00:00:51.795 --> 00:00:55.058
Now problem 34.

00:00:55.058 --> 00:00:56.990
Let me copy and paste
this one.

00:01:01.706 --> 00:01:04.150
Let's see, I can paste it.

00:01:07.320 --> 00:01:11.410
All right, and they ask us,
which graph represents the

00:01:11.410 --> 00:01:15.420
solution to this system
of inequalities?

00:01:15.420 --> 00:01:17.900
OK, so we could take each of
these inequalities and then,

00:01:17.900 --> 00:01:21.300
what we want to do is we want
the area that satisfies both

00:01:21.300 --> 00:01:22.110
of these inequalities.

00:01:22.110 --> 00:01:24.390
So let's take a look
at the first one.

00:01:24.390 --> 00:01:27.140
And I like to always have it
in kind of y is equal to mx

00:01:27.140 --> 00:01:27.850
plus b form.

00:01:27.850 --> 00:01:28.530
So let's do that.

00:01:28.530 --> 00:01:32.530
So the first one says 2x
is greater than or

00:01:32.530 --> 00:01:35.360
equal to y minus 1.

00:01:35.360 --> 00:01:41.640
If we add 1 to both sides, we
get 2x plus 1 is greater than

00:01:41.640 --> 00:01:42.970
or equal to y.

00:01:42.970 --> 00:01:44.920
And if we just want to write
that in a way that we're used

00:01:44.920 --> 00:01:47.940
to seeing it, y-- I'm just
switching it around.

00:01:47.940 --> 00:01:52.600
y is less than or equal
to 2x plus 1.

00:01:52.600 --> 00:01:54.780
So let's see which one
this represents.

00:01:54.780 --> 00:01:57.980
We have a line, 2x plus 1, so
the y-intercept is 1 and the

00:01:57.980 --> 00:02:00.170
slope is 2.

00:02:00.170 --> 00:02:01.200
So let's see.

00:02:01.200 --> 00:02:03.300
So that's this line here.

00:02:03.300 --> 00:02:05.850
y-intercept 1 and if
we go over to the

00:02:05.850 --> 00:02:11.240
right 2, we go up 4.

00:02:11.240 --> 00:02:13.700
However much we increase
in x, we increase twice

00:02:13.700 --> 00:02:14.550
as much with y.

00:02:14.550 --> 00:02:16.400
That's what the slope
tells us.

00:02:16.400 --> 00:02:21.430
And that's this line in
this chart like that.

00:02:21.430 --> 00:02:21.920
Fair enough.

00:02:21.920 --> 00:02:22.720
So that's that line.

00:02:22.720 --> 00:02:24.460
And they're saying y
is less than that.

00:02:24.460 --> 00:02:27.930
So we want the area
below that graph.

00:02:27.930 --> 00:02:29.140
So it's not this one.

00:02:29.140 --> 00:02:32.500
This one they're shading in
the area above that line.

00:02:32.500 --> 00:02:36.030
For any x and y that's on the
line, they're shading in y is

00:02:36.030 --> 00:02:37.730
greater than that point.

00:02:37.730 --> 00:02:41.410
No, but we what y is less
than that point.

00:02:41.410 --> 00:02:44.220
For any y on the point,
all the y's less

00:02:44.220 --> 00:02:45.680
than it satisfy it.

00:02:45.680 --> 00:02:47.920
So based on the information now,
this one seems to be a

00:02:47.920 --> 00:02:48.785
pretty good candidate.

00:02:48.785 --> 00:02:52.160
I haven't looked at the
other line yet.

00:02:52.160 --> 00:02:55.950
And frankly, neither of--
I guess, no neither.

00:02:55.950 --> 00:02:58.060
Maybe this one might be a
candidate because this

00:02:58.060 --> 00:02:59.040
is also less than.

00:02:59.040 --> 00:03:02.610
Maybe we have to be less than
both of these lines, right?

00:03:02.610 --> 00:03:04.280
This is the line that
we just figured out.

00:03:04.280 --> 00:03:05.410
We're definitely less than it.

00:03:05.410 --> 00:03:08.210
Maybe we have to be less than
that second line as well.

00:03:08.210 --> 00:03:10.370
So if we have to be greater
than that second line, we

00:03:10.370 --> 00:03:11.260
haven't looked at that yet.

00:03:11.260 --> 00:03:13.220
It's going to be choice C.

00:03:13.220 --> 00:03:16.280
If we have to be less than the
second line and this first

00:03:16.280 --> 00:03:19.250
line, then we're going
to be in choice D.

00:03:19.250 --> 00:03:22.320
Let's see what the second piece
of information tells us.

00:03:22.320 --> 00:03:25.040
OK, well they actually have a
typo and they don't give us

00:03:25.040 --> 00:03:30.140
enough information
to actually-- let

00:03:30.140 --> 00:03:32.140
me look up the answer.

00:03:32.140 --> 00:03:34.510
Because they didn't even write
a greater than or equal or

00:03:34.510 --> 00:03:35.810
nothing here.

00:03:35.810 --> 00:03:39.330
So I have to see what they
probably intended.

00:03:39.330 --> 00:03:41.730
OK, so I just looked up the
answer and they said that the

00:03:41.730 --> 00:03:43.220
answer is C.

00:03:43.220 --> 00:03:46.780
So they must want us to be
greater than this second line.

00:03:49.460 --> 00:03:53.880
So let's figure out what
this should have been.

00:03:53.880 --> 00:03:56.000
What should have this-- should
there have been an equals?

00:03:56.000 --> 00:03:57.270
Well, definitely not
an equal sign.

00:03:57.270 --> 00:03:58.820
But should there have been
a greater than or

00:03:58.820 --> 00:03:59.800
less than or whatever?

00:03:59.800 --> 00:04:01.290
We'll figure that out.

00:04:01.290 --> 00:04:01.800
So let's see.

00:04:01.800 --> 00:04:08.270
2x minus 5y circle 10.

00:04:08.270 --> 00:04:09.930
We don't know if there's an
equal there or less-- this is

00:04:09.930 --> 00:04:12.545
almost a better exercise than
what they intended.

00:04:15.510 --> 00:04:15.890
Let's see.

00:04:15.890 --> 00:04:19.504
If we were to add 5y to both
sides, I just want to get y on

00:04:19.504 --> 00:04:20.040
the other side.

00:04:20.040 --> 00:04:21.959
And I don't want to have
to say flip signs and

00:04:21.959 --> 00:04:23.530
divide by minus 5.

00:04:23.530 --> 00:04:24.820
So let me get 5y on
the other side.

00:04:24.820 --> 00:04:26.140
So I'll add 5y to both sides.

00:04:26.140 --> 00:04:28.010
So we get 2x.

00:04:28.010 --> 00:04:31.420
Whatever inequality sign,
so that could

00:04:31.420 --> 00:04:33.330
be an equal or whatever.

00:04:33.330 --> 00:04:37.570
10 plus 5y.

00:04:37.570 --> 00:04:39.190
Subtract 10 from both sides.

00:04:39.190 --> 00:04:44.180
So you get 2x minus
10 circle 5y.

00:04:44.180 --> 00:04:46.210
Remember that circle could
be an equal sign.

00:04:46.210 --> 00:04:46.840
Well, it's not an equal.

00:04:46.840 --> 00:04:48.330
It's either a greater than
or equal or it's a

00:04:48.330 --> 00:04:49.650
less than or equal.

00:04:49.650 --> 00:04:56.400
And then we can divide both
sides by 5 and you get 2

00:04:56.400 --> 00:05:03.370
divided by 5 x minus 2, some
greater than or equal or less

00:05:03.370 --> 00:05:06.060
than or equal sign, y.

00:05:06.060 --> 00:05:08.690
Now they told us C was the
answer because I had to look

00:05:08.690 --> 00:05:11.990
it up because they didn't give
us any inequality sign there.

00:05:11.990 --> 00:05:14.460
If C is the answer that means
we're going to-- we want all

00:05:14.460 --> 00:05:17.410
y's that are below
the first line.

00:05:17.410 --> 00:05:20.330
So below this line right here.

00:05:20.330 --> 00:05:22.850
So that's that area.

00:05:22.850 --> 00:05:26.600
And if we're in this grey area,
all y's that are above

00:05:26.600 --> 00:05:30.040
this bottom line.

00:05:30.040 --> 00:05:32.400
So if we're above this bottom
line, this bottom line

00:05:32.400 --> 00:05:34.560
is 2/5 x minus 2.

00:05:34.560 --> 00:05:36.920
So we want all y's that
are greater than that.

00:05:40.180 --> 00:05:45.110
Sorry, I've done it-- we want
all the y's that are greater

00:05:45.110 --> 00:05:47.940
than that. y is greater
than this thing.

00:05:47.940 --> 00:05:50.810
So that sign in this
problem should have

00:05:50.810 --> 00:05:54.110
been a less than sign.

00:05:54.110 --> 00:05:56.570
I'm saying y is greater
than this.

00:05:56.570 --> 00:05:59.640
But if you read it left to
right, you have 2x minus 2 is

00:05:59.640 --> 00:06:01.090
less than y.

00:06:01.090 --> 00:06:02.400
And so we figured
out the sign.

00:06:02.400 --> 00:06:03.280
And you might be saying,
hey, wait.

00:06:03.280 --> 00:06:06.790
How come this area over
here doesn't work?

00:06:06.790 --> 00:06:08.650
Well if you think about
it, this area is

00:06:08.650 --> 00:06:11.080
above our first line.

00:06:11.080 --> 00:06:13.880
This area is y is greater
than 2x plus 1.

00:06:13.880 --> 00:06:15.670
It's above our first line.

00:06:15.670 --> 00:06:17.970
And it's actually below
our second line.

00:06:17.970 --> 00:06:19.530
So it's actually the
opposite area.

00:06:19.530 --> 00:06:21.420
Anyway, next problem.

00:06:21.420 --> 00:06:24.780
The answer was C.

00:06:24.780 --> 00:06:30.200
Next problem, 35.

00:06:30.200 --> 00:06:31.160
OK, let's see.

00:06:31.160 --> 00:06:32.510
I'll copy and paste this one.

00:06:32.510 --> 00:06:33.800
I think the choices
are interesting.

00:06:41.300 --> 00:06:44.340
What is the solution to this
system of equations?

00:06:44.340 --> 00:06:48.170
All right, so let's see if we
can get in a form that makes

00:06:48.170 --> 00:06:51.330
it easy to look at.

00:06:51.330 --> 00:06:52.770
Let's take that first
equation and let's

00:06:52.770 --> 00:06:54.640
add 3x to both sides.

00:06:54.640 --> 00:06:57.530
If you add 3x to both sides,
that first equation becomes

00:06:57.530 --> 00:07:03.990
positive 3x plus y is
equal to minus 2.

00:07:03.990 --> 00:07:06.440
All I did is I did plus 3--
you can't see that.

00:07:06.440 --> 00:07:11.080
I added plus 3x to both sides
of this equation.

00:07:11.080 --> 00:07:13.380
And of course, that
cancels with that.

00:07:13.380 --> 00:07:14.920
All right, and now let' see.

00:07:14.920 --> 00:07:20.740
That second equation is 6x plus
2 is equal to minus 2.

00:07:20.740 --> 00:07:22.810
Well let me do something else
to this equation and I think

00:07:22.810 --> 00:07:23.890
it'll become apparent
that these are

00:07:23.890 --> 00:07:25.650
actually the same line.

00:07:25.650 --> 00:07:28.510
So if you take this first
equation and multiply both

00:07:28.510 --> 00:07:30.460
sides of it by 2,
what do you get?

00:07:30.460 --> 00:07:31.160
Let's see.

00:07:31.160 --> 00:07:36.620
Multiply 3x plus y by 2, you
get 2 times 3x is 6x.

00:07:36.620 --> 00:07:37.990
Plus 2 times y.

00:07:37.990 --> 00:07:39.670
You have to distribute the 2.

00:07:39.670 --> 00:07:41.500
Plus 2y is equal to.

00:07:41.500 --> 00:07:45.650
Minus 2 times to 2 is minus 4.

00:07:45.650 --> 00:07:46.900
And they're the same line.

00:07:50.770 --> 00:07:54.370
So when you're solving a system
of equations, you're

00:07:54.370 --> 00:07:57.990
figuring out, where do those
two equations intersect?

00:07:57.990 --> 00:08:01.640
If they're the same lines, they
intersect everywhere.

00:08:01.640 --> 00:08:04.480
So they have an infinite
number of solutions.

00:08:04.480 --> 00:08:07.075
So the answer is choice D.

00:08:10.510 --> 00:08:11.760
Next problem.

00:08:14.650 --> 00:08:17.840
They give us another
one like that.

00:08:17.840 --> 00:08:20.180
They want to know the ordered
pair that's a

00:08:20.180 --> 00:08:22.040
solution to this equation.

00:08:22.040 --> 00:08:25.450
I'll just copy and paste
the equation there.

00:08:25.450 --> 00:08:28.830
OK, so the easiest thing to do
is probably to just subtract

00:08:28.830 --> 00:08:31.770
the second equation from
the first equation.

00:08:31.770 --> 00:08:33.690
And instead, I'll actually
write it out explicitly.

00:08:33.690 --> 00:08:39.110
So the first question is x
plus 3y is equal to 7.

00:08:39.110 --> 00:08:41.240
Instead of subtracting this one
from that one, let's just

00:08:41.240 --> 00:08:43.540
multiply the second question by
negative 1 and then we'll

00:08:43.540 --> 00:08:44.750
add the two equations.

00:08:44.750 --> 00:08:47.610
So if we multiply this bottom
equation by negative 1 we get

00:08:47.610 --> 00:08:52.660
minus x minus 2y is
equal to minus 10.

00:08:52.660 --> 00:08:54.310
And the whole reason why I'm
doing that is I know when I

00:08:54.310 --> 00:08:57.680
add these two left-hand sides,
the x and the minus x are

00:08:57.680 --> 00:08:58.570
going to cancel out.

00:08:58.570 --> 00:09:00.150
And then I can just
solve for y.

00:09:00.150 --> 00:09:02.510
That's why you immediately see,
they have an x and an x.

00:09:02.510 --> 00:09:07.610
If we subtract this from that,
they'll cancel out.

00:09:07.610 --> 00:09:10.850
So if we add these two
equations, the x' cancel out.

00:09:10.850 --> 00:09:14.510
3y minus 2y is equal to y.

00:09:14.510 --> 00:09:22.140
And 7 minus 10 is equal
to minus 3.

00:09:25.090 --> 00:09:25.560
Fair enough.

00:09:25.560 --> 00:09:27.500
And now we can substitute back
in to figure out an x.

00:09:27.500 --> 00:09:30.896
So let's use the
first equation.

00:09:30.896 --> 00:09:36.130
x plus 3 times minus 3-- we
figured out what y is-- is

00:09:36.130 --> 00:09:39.250
equal to 7.

00:09:39.250 --> 00:09:41.200
And then you get x.

00:09:41.200 --> 00:09:44.486
3 times-- minus 9
is equal to 7.

00:09:44.486 --> 00:09:47.620
x is equal to-- add 9 to both
sides of this equation.

00:09:47.620 --> 00:09:49.580
x is equal to 16.

00:09:49.580 --> 00:09:54.070
So the solution is 16 comma
minus 3. x and y.

00:09:54.070 --> 00:09:55.520
And that is choice D.

00:09:58.690 --> 00:10:08.860
All right, problem 37.

00:10:08.860 --> 00:10:11.850
Marcy has a total of 100
dimes and quarters.

00:10:11.850 --> 00:10:14.190
If the total value of the coins
is $14.05, how many

00:10:14.190 --> 00:10:15.780
quarters does she have?

00:10:15.780 --> 00:10:19.555
All right, so let's say that d
is a number of dimes and q is

00:10:19.555 --> 00:10:21.000
the number of quarters.

00:10:21.000 --> 00:10:24.290
So if you take the number of
dimes plus the number of

00:10:24.290 --> 00:10:28.900
quarters she has 100 coins.

00:10:28.900 --> 00:10:32.020
That's this piece of information
right there.

00:10:32.020 --> 00:10:35.510
And then the total value
of the coins.

00:10:35.510 --> 00:10:41.180
It's going to be 0.10 times the
number of dimes plus 0.25

00:10:41.180 --> 00:10:43.110
times the number of quarters.

00:10:43.110 --> 00:10:51.160
And they tell us that that
is equal to $14.05.

00:10:51.160 --> 00:10:52.410
And that's this piece
of information.

00:10:52.410 --> 00:10:54.760
The total value of the
coins is $14.05.

00:10:54.760 --> 00:10:56.960
And they say, well, how many
quarters does she have?

00:10:56.960 --> 00:10:59.640
So we just have to
solve for q.

00:10:59.640 --> 00:11:01.370
So let's do that.

00:11:01.370 --> 00:11:04.270
So if we want to cancel out the
d's, what we could do is

00:11:04.270 --> 00:11:06.640
we can multiply this
top equation by

00:11:06.640 --> 00:11:09.650
let's say, minus 0.1.

00:11:09.650 --> 00:11:11.510
And I'm doing that so
it cancels out with

00:11:11.510 --> 00:11:13.050
this d right here.

00:11:13.050 --> 00:11:14.850
Let me do it in a
different color.

00:11:14.850 --> 00:11:20.820
So if I multiply that top
equation times minus 0.1 I get

00:11:20.820 --> 00:11:27.500
minus 0.1d-- I could write
1-0 if I want-- d.

00:11:27.500 --> 00:11:35.030
Minus 0.10q is equal to-- what's
minus 1/10 of 100?

00:11:35.030 --> 00:11:38.650
Well that's minus 10, right?

00:11:38.650 --> 00:11:42.090
100 times 0.1 is 10 and we're
doing a minus 0.1.

00:11:42.090 --> 00:11:43.930
All right, now we can add
these two equations.

00:11:46.810 --> 00:11:50.210
0.1d minus 0.1d, those
cancel out.

00:11:50.210 --> 00:11:55.510
0.25q minus 0.1q, that's equal
to-- let me switch colors.

00:11:55.510 --> 00:12:01.030
That's equal to 0.15q
is equal to-- and

00:12:01.030 --> 00:12:03.550
what's 14.05 minus 10?

00:12:03.550 --> 00:12:07.230
That's equal to $4.05.

00:12:07.230 --> 00:12:09.220
Just to get rid of the decimals,
we can multiply both

00:12:09.220 --> 00:12:12.860
sides of this equation by 100.

00:12:12.860 --> 00:12:20.160
So we'll get 15q is
equal to 405.

00:12:20.160 --> 00:12:20.600
So q.

00:12:20.600 --> 00:12:22.760
We divide both sides
of this by 15.

00:12:22.760 --> 00:12:27.480
And so how many times
does 15 go into 405?

00:12:27.480 --> 00:12:28.870
15 goes into 40, what?

00:12:28.870 --> 00:12:29.570
Two times.

00:12:29.570 --> 00:12:34.940
2 times 15 is 30.

00:12:34.940 --> 00:12:36.080
Get a 10.

00:12:36.080 --> 00:12:37.310
105.

00:12:37.310 --> 00:12:41.470
15 goes into 105 seven
times I think.

00:12:41.470 --> 00:12:44.540
7 times 5 is 35.

00:12:44.540 --> 00:12:45.750
7 times 1 is 7.

00:12:45.750 --> 00:12:46.990
Plus 3 is 10.

00:12:46.990 --> 00:12:47.670
27.

00:12:47.670 --> 00:12:50.100
So q is equal to 27.

00:12:50.100 --> 00:12:53.570
So Marcy has 27 quarters.

00:12:53.570 --> 00:12:56.030
Anyway, see you in
the next video.


CA Algebra I: Simplying Expressions

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.280
We're on problem 38.

00:00:02.280 --> 00:00:05.040
Which of the following best
describes the graph of this

00:00:05.040 --> 00:00:07.410
system of equations?

00:00:07.410 --> 00:00:09.130
OK, so maybe they're
the same line.

00:00:09.130 --> 00:00:10.200
Maybe they're parallel.

00:00:10.200 --> 00:00:12.240
Maybe they only intersect
in one point-- two lines

00:00:12.240 --> 00:00:14.930
intersecting in only
two points.

00:00:14.930 --> 00:00:16.590
Well that's impossible.

00:00:16.590 --> 00:00:19.170
Two lines, I mean that can
happen with curves, but that's

00:00:19.170 --> 00:00:20.220
not going to happen
with lines.

00:00:20.220 --> 00:00:22.460
So we can already cancel
out choice D.

00:00:22.460 --> 00:00:25.690
OK, now let's look
at these two.

00:00:25.690 --> 00:00:27.445
See I have a y here and
I have a 5y here.

00:00:27.445 --> 00:00:30.610
Let's multiply this top equation
times 5 and see what

00:00:30.610 --> 00:00:31.940
it looks like.

00:00:31.940 --> 00:00:33.660
So if you multiply the
left-hand side

00:00:33.660 --> 00:00:38.370
by 5, you get 5y.

00:00:38.370 --> 00:00:39.200
I'll do it up here.

00:00:39.200 --> 00:00:47.750
You get 5y is equal to-- 5 times
minus 2 is minus 10x,

00:00:47.750 --> 00:00:50.980
plus 5 times 3 is 15.

00:00:50.980 --> 00:00:53.310
So if you multiply the top
equation-- both sides of it--

00:00:53.310 --> 00:00:56.390
by 5 and it really doesn't
fundamentally change the line,

00:00:56.390 --> 00:00:59.230
the equation might look
different, but the equality

00:00:59.230 --> 00:01:01.790
will still hold in the same
universe, which is

00:01:01.790 --> 00:01:03.470
essentially that line.

00:01:03.470 --> 00:01:05.630
So if you just multiply both
sides by 5, they become the

00:01:05.630 --> 00:01:06.790
same equation.

00:01:06.790 --> 00:01:09.370
5y is equal to minus
10x plus 15.

00:01:09.370 --> 00:01:11.570
So they are the same lines.

00:01:11.570 --> 00:01:14.010
So that's A, two identical
lines.

00:01:16.570 --> 00:01:21.270
Problem 39.

00:01:21.270 --> 00:01:28.960
And they want us to simplify
5x to the third over 10x to

00:01:28.960 --> 00:01:30.740
the seventh.

00:01:30.740 --> 00:01:33.560
So the easiest way to think of
this, or at least for me--

00:01:33.560 --> 00:01:34.730
well, there's a lot of ways
you can do it and

00:01:34.730 --> 00:01:35.550
we'll do it both ways.

00:01:35.550 --> 00:01:42.610
This is the same thing as 5/10
times x to the third times x

00:01:42.610 --> 00:01:45.010
to the minus 7.

00:01:45.010 --> 00:01:48.640
1 over x to the 7 is the same
thing as x to the minus 7.

00:01:48.640 --> 00:01:53.000
And this is equal to--
5/10 is 1/2.

00:01:53.000 --> 00:01:54.790
And then here, we have the
same base and we're

00:01:54.790 --> 00:01:56.920
multiplying, so we can
add the exponents.

00:01:56.920 --> 00:01:59.640
3 plus minus 7 is minus 4.

00:01:59.640 --> 00:02:02.650
So x to the minus
fourth power.

00:02:02.650 --> 00:02:06.690
And we could write that as 1/2
times 1 over x to the fourth

00:02:06.690 --> 00:02:10.030
or 1 over 2x to the fourth.

00:02:10.030 --> 00:02:12.790
And that is choice B.

00:02:12.790 --> 00:02:13.650
Now you could have done
it other ways.

00:02:13.650 --> 00:02:17.090
You could have said,
OK, let's see.

00:02:17.090 --> 00:02:19.500
Divide the numerator and
denominator by 5.

00:02:19.500 --> 00:02:21.030
So this would have been 1.

00:02:21.030 --> 00:02:21.960
This would have been a 2.

00:02:21.960 --> 00:02:24.170
And you say, OK, let's divide
the numerator and denominator

00:02:24.170 --> 00:02:25.780
by x to the third.

00:02:25.780 --> 00:02:27.350
So this will become a 1.

00:02:27.350 --> 00:02:29.190
And x to the seventh
divided by x to the

00:02:29.190 --> 00:02:32.180
third is x to the fourth.

00:02:32.180 --> 00:02:32.840
You could have done
it that way.

00:02:32.840 --> 00:02:36.190
You had 1 over 2x
to the fourth.

00:02:36.190 --> 00:02:36.770
Either way.

00:02:36.770 --> 00:02:38.380
Or you could have even said--
you didn't have

00:02:38.380 --> 00:02:39.060
to go to this step.

00:02:39.060 --> 00:02:42.060
You could have said, OK, when
I'm dividing with the same

00:02:42.060 --> 00:02:44.370
base, I can just subtract
the exponents.

00:02:44.370 --> 00:02:47.500
So 3 minus 7 was minus 4.

00:02:47.500 --> 00:02:47.930
Either way.

00:02:47.930 --> 00:02:52.580
All of them would have valid
ways to approach this problem.

00:02:52.580 --> 00:02:55.870
Problem 40.

00:02:55.870 --> 00:02:58.130
This looks like a
simplification.

00:02:58.130 --> 00:03:08.200
They write 4x squared minus 2x
plus 8, minus x squared plus

00:03:08.200 --> 00:03:12.670
3x minus 2 is equal to.

00:03:12.670 --> 00:03:15.650
So the key here is just to
realize that this is a minus.

00:03:15.650 --> 00:03:18.650
So you could kind of view it as
a plus minus 1 times this

00:03:18.650 --> 00:03:19.430
whole thing.

00:03:19.430 --> 00:03:22.000
So we're just going to have
to distribute that out.

00:03:22.000 --> 00:03:29.890
So this is equal to 4x squared
minus 2x plus 8.

00:03:29.890 --> 00:03:32.090
And now we distribute this
minus over this whole

00:03:32.090 --> 00:03:32.600
expression.

00:03:32.600 --> 00:03:36.990
So minus times x squared
is minus x squared.

00:03:36.990 --> 00:03:39.750
Minus times 3x, positive 3x.

00:03:39.750 --> 00:03:42.250
So it's minus 3x.

00:03:42.250 --> 00:03:46.290
The minus 1 times negative 2.

00:03:46.290 --> 00:03:49.100
Well now they cancel out
and you get a plus 2.

00:03:49.100 --> 00:03:50.610
We switch the sign on everything
here because

00:03:50.610 --> 00:03:53.790
they're all being multiplied
by this negative 1.

00:03:53.790 --> 00:03:55.470
OK, now we can simplify.

00:03:55.470 --> 00:03:58.840
So let's take the x squared
terms first. So we have a 4x

00:03:58.840 --> 00:04:01.410
squared, we have a
minus x squared.

00:04:01.410 --> 00:04:05.400
So 4x squared minus x squared
is 3x squared.

00:04:05.400 --> 00:04:07.480
4 minus 1 is 3.

00:04:07.480 --> 00:04:11.920
Then let's do our x terms.
We have a minus 2x, we

00:04:11.920 --> 00:04:13.490
have a minus 3x.

00:04:13.490 --> 00:04:16.209
So minus 2 minus 3,
that's a minus 5x.

00:04:19.490 --> 00:04:20.679
And then last we have
our constants.

00:04:20.679 --> 00:04:23.080
We have 8 plus 2.

00:04:23.080 --> 00:04:25.150
8 plus 2 is 10.

00:04:25.150 --> 00:04:29.340
So 3x squared minus
5x plus 10.

00:04:29.340 --> 00:04:32.850
And that is choice D.

00:04:32.850 --> 00:04:37.450
Problem 41.

00:04:37.450 --> 00:04:38.700
OK.

00:04:41.400 --> 00:04:44.010
They say the sum of
two binomial-- let

00:04:44.010 --> 00:04:46.420
me copy this one.

00:04:46.420 --> 00:04:47.670
It's interesting.

00:04:50.510 --> 00:04:54.840
The sum of two binomials
is 5x squared minus 6x.

00:04:54.840 --> 00:04:57.810
So a binomial is just a
polynomial with two terms. If

00:04:57.810 --> 00:05:01.660
one of the binomials is 3x
squared minus 2x, what is the

00:05:01.660 --> 00:05:02.910
other binomial?

00:05:04.880 --> 00:05:07.790
So this binomial is one of them,
so they're saying 3x

00:05:07.790 --> 00:05:13.430
squared minus 2x, and when you
add that to some other

00:05:13.430 --> 00:05:18.960
binomial-- and I don't know, let
me just write that as A.

00:05:18.960 --> 00:05:22.290
I mean there is no constant
term here and there is no

00:05:22.290 --> 00:05:24.940
constant term here, so I'm
assuming that my-- and it has

00:05:24.940 --> 00:05:25.860
to be a binomial.

00:05:25.860 --> 00:05:29.600
There's only two terms. So I'm
assuming my two terms are an x

00:05:29.600 --> 00:05:31.900
squared term and x term because
that's the only terms

00:05:31.900 --> 00:05:33.870
that are involved in
both of these.

00:05:33.870 --> 00:05:38.770
So let's say my binomial
is Ax squared plus Bx.

00:05:38.770 --> 00:05:40.590
This is the mystery binomial.

00:05:40.590 --> 00:05:43.390
And their sum is equal
to this up here.

00:05:43.390 --> 00:05:49.360
Is equal to 5x squared
minus 6x.

00:05:49.360 --> 00:05:51.110
Now let's see what we can do.

00:05:51.110 --> 00:05:52.550
Well this is a plus here,
so the parenthesis

00:05:52.550 --> 00:05:53.230
really don't matter.

00:05:53.230 --> 00:06:03.420
We can rearrange this as 3x
squared plus Ax squared minus

00:06:03.420 --> 00:06:11.780
2x plus Bx is equal to
5x squared minus 6x.

00:06:11.780 --> 00:06:13.920
3 plus A.

00:06:13.920 --> 00:06:17.400
3x squared plus Ax squared,
that's the same thing as 3

00:06:17.400 --> 00:06:21.270
plus A, x squared.

00:06:21.270 --> 00:06:25.750
And then, minus 2x plus Bx, or
we could switch them around.

00:06:25.750 --> 00:06:30.490
That's the same thing as plus
B minus 2-- I just took the

00:06:30.490 --> 00:06:33.220
coefficients and added
them together-- x.

00:06:33.220 --> 00:06:35.160
I switched them, but we could
have written this in the other

00:06:35.160 --> 00:06:41.560
order to begin with-- is equal
to 5x squared minus 6x.

00:06:41.560 --> 00:06:42.480
And now you just compare.

00:06:42.480 --> 00:06:45.530
OK, 3 plus A-- if you just look
at the x squared terms--

00:06:45.530 --> 00:06:48.720
3 plus A has to be equal to 5.

00:06:48.720 --> 00:06:51.190
Because that's the coefficient
on the x squared term.

00:06:51.190 --> 00:06:54.380
So 3 plus A is equal to 5.

00:06:54.380 --> 00:06:55.560
Subtract 3 from both sides.

00:06:55.560 --> 00:06:57.630
You get A is equal to 2.

00:06:57.630 --> 00:07:03.200
And then we have B minus 2 has
to be the coefficient on x

00:07:03.200 --> 00:07:05.980
here, so it has to be
equal to minus 6.

00:07:05.980 --> 00:07:08.640
Add 2 to both sides,
you get B.

00:07:08.640 --> 00:07:11.130
Minus 6 plus 2 is 4.

00:07:11.130 --> 00:07:13.600
So the other binomial, just
substituting that Ax squared

00:07:13.600 --> 00:07:20.540
plus Bx, is 2x squared
plus Bx.

00:07:20.540 --> 00:07:20.910
Oh, sorry.

00:07:20.910 --> 00:07:22.450
This is a minus 4.

00:07:22.450 --> 00:07:25.060
Minus 6 plus 2 is minus 4.

00:07:25.060 --> 00:07:26.230
So plus Bx.

00:07:26.230 --> 00:07:30.790
So minus 4-- that's B-- x.

00:07:30.790 --> 00:07:33.680
And that is choice A.

00:07:33.680 --> 00:07:34.930
Next problem.

00:07:41.380 --> 00:07:44.000
OK, they say, which of the
following expressions is equal

00:07:44.000 --> 00:07:45.980
to-- this is problem 42.

00:07:45.980 --> 00:07:56.400
And they write x plus 2, plus
x minus 2, times 2x plus 1.

00:07:56.400 --> 00:07:57.590
So we have to simplify this.

00:07:57.590 --> 00:07:59.750
And remember, order of
operations, multiplication

00:07:59.750 --> 00:08:01.720
comes first. So we have to
multiply these two expressions

00:08:01.720 --> 00:08:04.410
first. So let's do that.

00:08:04.410 --> 00:08:06.900
So this is-- I'll rewrite
this one over here.

00:08:06.900 --> 00:08:12.410
x plus 2 plus-- and now
let's multiply this.

00:08:12.410 --> 00:08:14.800
When you multiply these two
binomials, you're really just

00:08:14.800 --> 00:08:16.860
doing the distributive
property twice.

00:08:16.860 --> 00:08:19.230
And let me show that to you.

00:08:19.230 --> 00:08:22.740
We could view this as x minus
2 times 2x plus x

00:08:22.740 --> 00:08:23.970
minus 2 plus 1.

00:08:23.970 --> 00:08:27.060
So I'm just distributing the x
minus 2 times each of these

00:08:27.060 --> 00:08:40.340
terms. So I could write this as
x minus 2 times 2x, plus x

00:08:40.340 --> 00:08:44.600
minus 2 times the 1.

00:08:44.600 --> 00:08:46.880
All right, and now we can just
simplify that by doing the

00:08:46.880 --> 00:08:48.300
distributive property again.

00:08:48.300 --> 00:08:53.840
So this is x plus 2 plus-- let's
distribute the 2x times

00:08:53.840 --> 00:08:54.290
each of these.

00:08:54.290 --> 00:08:58.200
2x times x is 2x squared.

00:08:58.200 --> 00:09:02.060
2x times minus 2 is minus 4x.

00:09:02.060 --> 00:09:03.680
Plus, well, we're
disturbing a 1.

00:09:03.680 --> 00:09:04.970
1 times anything
is just itself.

00:09:04.970 --> 00:09:06.700
So plus x minus 2.

00:09:10.430 --> 00:09:11.730
And let's see what we can do.

00:09:11.730 --> 00:09:15.240
We only have 1x squared terms,
so let's write that down.

00:09:15.240 --> 00:09:17.420
2x squared.

00:09:17.420 --> 00:09:20.560
So 2x squared.

00:09:20.560 --> 00:09:25.440
And then our x terms, we
have a plus x, a minus

00:09:25.440 --> 00:09:27.530
4x, and a plus x.

00:09:27.530 --> 00:09:30.430
So we have 1 minus
4 is minus 3.

00:09:30.430 --> 00:09:32.130
Plus 1 is minus 2.

00:09:32.130 --> 00:09:34.840
So its minus 2x.

00:09:34.840 --> 00:09:35.920
And then, let's see.

00:09:35.920 --> 00:09:37.480
We have a positive
2 and a minus 2.

00:09:37.480 --> 00:09:39.370
They cancel out.

00:09:39.370 --> 00:09:44.690
So we're left with 2x squared
minus 2x, and that's choice A.

00:09:49.340 --> 00:09:53.050
Problem 43, I think we
can fit in here.

00:09:53.050 --> 00:09:54.300
Let me copy and paste it.

00:09:57.290 --> 00:10:02.160
OK, copy it and now
pasting it.

00:10:02.160 --> 00:10:04.960
OK, it says, a volleyball
court is

00:10:04.960 --> 00:10:06.490
shaped like a rectangle.

00:10:06.490 --> 00:10:09.670
Let me draw that.

00:10:09.670 --> 00:10:11.490
Well, I didn't want to draw
it filled in like

00:10:11.490 --> 00:10:13.740
that, but fair enough.

00:10:13.740 --> 00:10:14.870
Shaped like a rectangle.

00:10:14.870 --> 00:10:18.970
It has a width of x meters and
a length of 2x meters.

00:10:18.970 --> 00:10:21.140
So it's width is x.

00:10:21.140 --> 00:10:25.060
Let me write, this could be
x and this would be 2x.

00:10:25.060 --> 00:10:26.330
Because this is longer.

00:10:26.330 --> 00:10:28.520
Which expression gives
the area of the

00:10:28.520 --> 00:10:29.660
court in square meters?

00:10:29.660 --> 00:10:32.680
Well the area is just the
width times the length.

00:10:32.680 --> 00:10:40.510
So it's just x times 2x, which
is equal to 2x squared.

00:10:40.510 --> 00:10:44.080
That's the same thing as 2
times, x times x, which is the

00:10:44.080 --> 00:10:45.490
same thing as 2x squared.

00:10:45.490 --> 00:10:47.720
And that's choice B.

00:10:47.720 --> 00:10:49.890
Anyway, see you in
the next video.