Section 9.5 is about solving quadratic equations
using the quadratic formula.
There are lots of ways to solve quadratics,
we will use the quadratic formula.
You will not be given this formula on your
tests so you are going to want to make sure
that you memorize it.
Our
first quadratic, x squared plus 8x plus 15
equals 0.
We are going to solve using the quadratic
formula.
The quadratic formula 
is x equals negative b plus or minus the square
root of b squared minus 4ac all over 2a.
What
this does, is this gives us our solutions
to the equation.
The place where our graph crosses the x-axis
when y = 0.
Now, you want to make sure that your quadratic
equation is in standard form.
This one is in standard
form the next one we do will not be in standard
form.
Lead off with your x squared term and then
you should have your term that has an x in
it next and then your constant and it should
be equal to 0.
Then substitute in.
Standard form is ax squared plus bx plus c.
So the constant is c, the coefficient of x
is b and the coefficient of x squared is a.
So, the negative b means the opposite of b
since b is positive 8 here, the opposite of
b is negative 8 plus or minus the square root
of b squared so 8 squared minus 4 times a
(x squared has a coefficient of 1) times c
all over two times a.
And then we need to simplify.
Negative 8 plus or minus the square root of
64 minus 60 all over 2.
x equals negative 8 plus or minus the square
root of four all over 2.
x equals negative 8 plus or minus
2 because the square root of four is 2, all
over 2.
And we have two possible solutions here.
The first one, x = negative 8 plus 2 divided
by 2 so that's negative 6 divided
by 2 which is negative 3.
And then the second one, x equals negative
8 minus 2 divided by 2.
Negative 10 divided by 2 is negative 5.
So my two solutions are negative 5 and negative
3.
Now this quadratic is not in standard form.
So I am going to subtract 2x and subtract
7 from both sides.
That will put it in standard form.
4x squared minus 2x minus 7 equals 0.
a is 4, b is negative 2 (you always want to
take the sign that's in front of your coefficient
or your constant), and c is negative 7.
Now
let's use the quadratic formula x equals the
opposite of b.
Since b is negative 2, the opposite of b is
positive 2 plus or minus the square root of
negative 2 squared minus
4 times a which is 4 times c which is negative
7 all over two times a so two times four.
x equals 2 plus or minus the square root of
4 minus (four times four times negative 7
is negative 112) so minus negative 112 all
over 8.
x is equal to 2 plus or minus the square root
of 116 all over 8.
And then 116 is not a perfect square so you
may be asked
to give your answer in exact form.
If you are asked for exact form you'll leave
your answer like this: 2 plus or minus the
square root of 116 divided by 8 equals x.
If you
are asked to approximate you will give a decimal.
So 2 plus or minus 10.77 divided by 8.
And then we have the same set up here.
2 plus 10.77 divided by 8 will give us one
solution.
So that's 12.77 divided by 8 which is 1.596
so we will say 1.60 and then 2 minus 10.77
divided by 8 is and then 2 minus 10.77 divided
by 8 is negative 1.096 so negative 1.10.
And so if I am asked for an approximate solution,
-1.10 and 1.60 will be the two solutions.
