hi class today we're going to use the quadratic
formula to solve quadratic equations we're
going to go through a couple examples and
at the end i'm going to derive the quadratic
equation okay the general quadratic equation
is we have ax^2 + bx + c = 0 for example we
could have x^2 + 5x + 6 = 0 here a is 1 here
b is 5 and here c is 3 for this general equation
to solve the quadratic formula is just -b
plus or minus the square root of b^2 - 4 * a
* c all over 2 * a this is the quadratic formula
okay and it solves every single quadratic
equation so let's do a couple examples let's
say we have okay so let's show an example
we have x^2 + 2x - 15 = 0 now to solve this
the first thing we should do is see if you
can factor it and i'll use an example here
that does factor and so we can solve it that
way and i'm going to use the quadratic equation
to show that we get the same answers then
we'll use the quadratic formula to solve ones
that we can't factor but this factors into
x well there's 3 * 5 goes here and the difference
is 2 and there's more positives so x + 5 times
x - 3 = 0 so we have x + 5 = 0 if A * B = 0
either A = 0 or B = 0 or x - 3 = 0 and we
have x = -5 there's a solution and we have
x = 3 there's a solution now let's use the
quadratic equation okay or the quadratic formula
all quadratic equations look like this plus
bx plus c = 0 so in this case a = 1 b is equal
to this guy here and c = -15 and the quadratic
formula is just -b plus or minus the square
root of b^2 - 4 * A * C all over 2 * A and
that's what our values of x are equal to so
we can plug those in we get x = -b so -2 plus
or minus the square root of b^2 a 4 minus
4 times a which is 1 times c which is a -15
all that over 2 * A or 2 * 1 okay so x = -2
plus or minus the square root of well what
do we have here we have 4 * 1 is 4 4 * 15
is 60 and the negatives become positives 4
+ 60 or 64 all over 2 so that's equal to the
square root of 64 is 8 so we have -2 plus
or minus 8 over 2 okay so one answers x = -2
+ 8 over 2 what's that equal to 6/2 which
is equal to 3 and that's one of the answers
we got right here and the other answer is
equal to negative 2 - 8 all over 2 and that's
equal to -2 and that's equal to -10/2 which
is -5 right and that's the other answer we
got here so you can see that the quadratic
equation works and can even work for a situation
that we can factor though factoring is much
quicker so if you can factor first see if
you can factor and if not you can use the
quadratic equation so let's look at another
example this example is x^2 + 3x - 11 = 0
now if you look at that you can't factor it
so we're going to use the quadratic formula
a = 1 comma b = 3 and c = -11 so we have -b
plus or minus the square root of b^2 - 4 * A
* C all over 2 * A right so our x values are
equal to these and so we have a -3 plus or
minus the square root of b^2 or 9 minus four
times a * c a -11 all over 2 * 1 so x = -3
plus or minus the square root of let's see
that's 44 and 9 is 53 all over 2 so there's
our answers one is x = -3 plus the square
root of 53 over 2 and another answer is is
a -3 minus the square root of 53 over 2 and
you can plug that into your calculator and
get values but it's often written like this
and let me just write it a bit clearer -3
plus or minus the square root of 53 all over
2 and you can write it like this and that's
understood to be these two answers because
of the plus or minus and that really does
give us two answers okay we can't factor that
and that's why we're getting a square root
that doesn't simply to a nice integer alrighty
let's try another example and we'll do a different
one 4x^2 this A term is not equal to 1 minus
3x + 6 = 0 so a = 4 b = -3 and c =6 so -b
plus or minus the square root of b^2 - 4AC
all over 2A = we have a negative negative
3 so it's a positive 3 plus or minus square
root of b^2 which is 9 minus 4 * A minus 4
* 4 * c 6 all over 2a which is 2 * 4 or 8
okay so our x values are equal to this stuff
alright so x = a 3 plus or minus well what
do we have here we have 9 - 4 * 4 * 6 so okay
we have square root of 9 - 96 okay all over
8 now this is where we get into a second type
of answer we have 3 plus or minus the square
root of a negative 87 right all over 8 well
you can't take the square root of -87 not
in real but we can imaginary numbers remember
i^2 = -1 right so we can factor that out we
have an i so x = 3 plus or minus i * the square
root of 87 all over 8 and actually it's more
typically written as 3 plus or minus the square
root of 87 i i is on the outside of that square
root all over 8 okay so there's really three
types if this b^2 i'm going to go to a new
page here there's really three types of solutions
we have and it's called a discriminant 
right we have -b plus or minus the square
root of b^2 - 4AC all over 2A right it all
depends on this b^2 - 4AC right so if b^2
- 4AC is greater than 0 then we have a square
root of a positive number and we get 2 real
solutions okay if b^2 - 4AC is equal to 0
then we get 1 real solution okay because that's
just 0 then so we just have -b over 2a as
a solution then we can also have -4ac is less
than 0 as we saw in our last example and then
we have a square root of a negative number
so we use i so we have 2 imaginary 
solutions 
okay so let's see that in action 
okay so let's see if we have x^2 + 8x + 3
right so what is b^2 - 4ac is going to be
equal to b is 8 so 64 - 4 * 4a which is 1
times c wihch is 3 and that's 64 - 12 that's
greater than 0 so we have two real solutions
okay and say we have 9x^2 - 12x + 4 = 0 then
b^2 - 4ac is going to be equal to well we
have b = -12 and squared is 144 minus 4 times
a which is 9 times c which is 4 okay 
and 16 times 9 is 144 as well and that's equal
to 0 so we only have one real solution and
let's get the solution x = -b plus or minus
the square root of b^2 - ac all over 2a is
just going to be equal to well b^2 - 4ac is
0 so that's 0 which is going to be equal to
a -b over 2a which is equal to a positive
12 minus a negative 12 is 12 over 2 * a a
is 9 so that's 18 so then we have one real
answer which is 2/3 okay just to solve that
one give you an example we get one real solution
so one real solution 
and if we go back to slides to this one here
if we look at our discriminant 
okay on that one that was 4x^2 - 3x + 6 was
equal to 0 this is example a b and c okay
and b^2 - 4ac what's equal to b^2 is 9 negative
times a negative is positive minus 4 * a which
is 4 times c which is 6 and that's clearly
9 - a pretty big number that's less than 0
so we get 2 imaginary solutions as we saw
okay so there we go okay now we can derive
the quadratic equation if we have ax^2 + bx
+ c is equal to 0 what we need to do is complete
the square the way we do that is we divide
both sides by a so i get x^2 + b over a x
+ c/a = 0/a which is just 0 now how do we
complete the square we take half of this we
take b over 2a and we square that remember
from completing the square so we have b^2
over 4a^2 to complete the square so we have
to get that on this side so i'll first subtract
the c/a so we get x^2 + b/a x is equal to
a -c over a okay now we add b^2 over 4a^2
and b^2 over 4a^2 to both sides right now
this is a perfect square this is x + b over
2a squared is equal to well let's just get
a common denominator multiply top and bottom
by well the 4a so 4a and 4a so if we combine
those we get b^2 - 4ac and that's all over
4a^2 and you can just get a common denominator
there and put the negative there and now we
take the square root function and square root
property so b a plus b over 2a is equal to
the square root of this will be plus or minus
the square root of b^2 - 4ac all over 4a^2
now you can see it i'm going to then add or
subtract b over 2a over here and we get x
= -b over 2a plus or 
minus well the square root of 4a^2 is just
2a and then we have ontop here b^2 - 4ac and
we have a common denominator so we can combine
those so -b plus or minus the square root
b^2 - 4ac all over 2a and there's the quadratic
formula derived from a general quadratic equation
right here well i hope you enjoyed this video
and it was helpful to your learning cheers
