>> This is Part 4 of solving
a quadratic equation using the
quadratic formula right
here and we're going
to solve this equation
using the quadratic formula
and also by factoring.
The first thing we want to do
with this is simplify each side
so we can get it
into standard form.
So we need to do
distributive property;
7X times X is 7X squared minus
7X times 2 is 14X plus 2X plus 8
and let's go ahead and
subtract 3 from both sides
so that we have it equal to 0.
So let's see if we combine
like terms here we have 7X
squared minus 12X plus 5
equals 0.
Now, to be honest
if you're trying
to solve a quadratic equation
and if you could
factor it that's going
to be usually the quickest way
to do it, so let's go ahead
and do this by factoring.
You have to see if it factors.
The only possibilities
here could be 7X and X
for the first terms and the only
possibilities here could be 5
and 1 so the question
is where would the 5 go
and where would the 1 go,
let's see if I do 5 here 1
there both minus that's the
correct factorization.
If you multiply that
back out you'll get back
to 7X squared minus
12X plus 5 equals 0.
So then we could just set each
factor equal to 0 and solve.
[ Silence ]
>> Divide both sides by
7 so we got X equals 5/7
and then we'll do this other one
X minus 1 equals 0 so X equals 1
so it's the other solution.
And, of course, the
only way you know
for sure these are correct is
you would have to plug each back
in the original equation
right here one at a time,
and that's not so awful to do
all you're going to have to deal
with getting common
denominators,
etcetera to add inside the
parenthesis and be very careful
with fractions at least with
5/7; 1 it'll be easy to check.
Okay, so let's now do the
problem using the quadratic
formula, so remember
we've got 5/7 and 1.
Alright, so this is
our original problem.
When we did it by factoring we
got 5/7 and 1 when we simplified
and put it in standard form we
have 7X squared minus 12X plus
5, so if we're going to use the
quadratic formula these are our
values for A, B, and C. Now,
instead of plugging them
in directly I'm going
to go ahead and figure
out what B squared minus 4AC
is so I know what's going
to go underneath that radical.
So I'm going to have B squared.
Well, if B is negative 12 B
squared will be negative 12
times negative 12, 144 minus --
now, it says 4AC so it's going
to be 4 times AC that's 7 times
5, which is 35, so that's 144
and I'm reading multiplication
over here first before
you subtract
so 4 times 35 is
140 and so that's 4.
So what this does is when
you're going to put it
in the quadratic formula now
you know that you're going
to have the square root of 4.
Remember it's the
B squared minus 4AC
that goes underneath
the radical,
so we've got a fraction bar.
Alright, so the quadratic
formula says the opposite E,
alright, so if B's negative 12
you're going to do the opposite
of B so it's 12 plus or minus
the square root of 4, right,
square root of this
number right here, right?
So what's the square root
of 4, 2, all over 2A.
Alright, well, what's
A, 7 so it's over 14.
So I'm showing just
sort of a little bit
of a shortcut here instead of
having to write everything out,
which it's also good
to be able to do that.
Alright, so this gives me 2
solutions here 12 plus 2 over 14
and 12 minus 2 over 14.
Alright, so 12 plus 2 over 14,
what's that, that's 14 over 14
or 1 and what's 12 minus 2 over
14 that's 10/14 which reduces
to 5/7 so notice I got the
same answers 1 and 5/7.
Alright let's just do
this problem one more time
where we don't do anything
in our head we show
everything all the steps
and then you've got your
choice what's easier
for you that's really key.
Alright here's the
original problem
after we simplified it here's
the values of A, B, and C,
here's the quadratic formula
and here you could see
where I'm plugging in
the values of A, B,
and C into the quadratic
formula.
So where there's a B I'm
putting in negative 12
where there's an
A I'm putting in 7
and where there's a
C I'm putting in 5.
Alright, so this is
what it would look like
and then we would simplify that,
so opposite of negative 12 is 12
and then you would just
simplify underneath the radical
at this point it's still going
to be 144 minus 140 over 14
so it's going to be 12 plus
or minus the square root of 4
over 14 which is 12
plus or minus 2 over 14
and there are your two
different solutions you're going
to get just like we
did it the other way
from here looks exactly the same
12 plus 2 over 14 or 12 minus 2
over 14 so you're going
to get 14 over 14 or 1
or 10 over 14 which is 5/7.
So you're going to get
your solution either way.
Let's go ahead and check 1
since that's the easy one
into the original equation.
Okay why don't you
put the video on pause
and try it on your own first.
Okay here we go we're
going to plug in 1 for X
so I have 7 times 1 times 1
minus 2 plus 2 times 1 plus 4.
Okay, so you see I'm
plugging in the 1
and so I have 7 times 1 that's 7
and 1 minus 2 is
negative 1 plus 2 times 5
so I have negative 7 plus
or 3 left side's 3 right
side's 3 and it checks.
I think I can get the check
for the next one in here
so let's go ahead
and check 5/7 too.
Alright, so we're
going to check 5/7.
It's good practice for you to
try plugging in 5/7 on your own
and seeing if this left
side simplifies to 3;
this is just practice
with fractions here.
Alright, so I'm plugging in
5/7 everywhere there's an X.
And inside parenthesis I'm
going to have to subtract
so I need a common denominator
so that's why I wrote 2 as 2
over 1 so I'm going to multiply
top and bottom here by 7 over 7
and the same thing in the
second parenthesis 7 over 7
so that my simplification's
going to be a little bit easier.
Okay, now, I have 7
times 5/7 I can go ahead
and cancel those 7's
so that I have 5 times
and then what's inside
parenthesis what's this going
to be.
I've got a common denominator
of 7 so I have 5 minus 14
which is negative 9, right,
so I've got negative 9/7 plus 2
times, alright, what do I have
over here I've got 7's and I've
got 5 plus 28 I think that's 33.
Alright, so I have 5 times
negative 9/7 that's negative
45/7 plus 66/7 because that's
2 times 33 and then I have
to do 66 minus 45 that's
21/7, woo hoo, 21/7 is 3.
So you know what, it
wasn't so bad after all.
It checked so we are now
certain that our solutions both
of them checked are 1 and 5/7.
Really this puts a lot of what
you learn in elementary algebra
and intermediate
algebra together
because you are solving
equations you're working
with fractions you could be
factoring all sorts of things.
Just practice, practice,
practice.
