Today we're going to talk about
solving and graphing 
quadratic equations.
Remember the zero property
of multiplication states that
if you multiply two things together
and the result is zero
one or both of them
have to be zero.
So formally that's written
A times B equals zero
and either A equals zero,
or B equals zero.
That's the first thing
we're going to use
to solve quadratic equations.
So if you have a quadratic equation,
remember if you were
to multiply these together,
you would have z squared plus 6z
minus 5z minus 30 equals zero.
So, z squared plus z
minus 30 equals zero.
That's a quadratic equation
because it has a squared term.
But, we're multiplying
two things together.
Remember this is factored
and z minus five is a factor
and z plus six is a factor.
z minus five is one number,
z plus six is another number,
multiplied together
they equal zero.
So either z minus five equals zero
or z plus six equals zero.
And now we can solve
a linear equation
because this is a exponent of one.
z minus five equals zero,
add 5 to both sides.
So z equals five
or z plus six equals zero,
subtract six from both sides,
so z equals negative six.
This is our solution.
And you are going to have
to check these in your workbook,
but I'm gonna leave that
for you to check if–
but I'll show you this one.
If we put five,
we have to do it separately.
So the first time
we're going to put five in.
Five minus five times five plus one
that's equal to zero times six,
which is zero.
Same thing with negative six minus five.
Negative six plus six
that's equal to negative 11 times zero,
which is zero.
So it checked out.
You can't put five in one
and six in the other,
or negative six in the other,
you have to do
all of your x's or z's as five
and then all of them 
as negative six.
So this one,
remember back here
I multiplied everything out
and got this,
but we really started from here.
This one we have to factor first
and then we can go back
and solve.
So, if we factor this,
this factors into x minus seven
times x minus two equals zero.
Remember you can use any
of your factoring tools that you want,
but for now I'm just going
to have it factored.
So that means that x minus seven
equals zero
or x minus two equals zero.
Add seven to both sides
of this equation
and we have x equals seven,
add two to both sides
of this equation
and we have x equals two.
So x equals seven
or x equals two.
Let's check our answer.
First we're going to check
x equals seven.
So we have...oops...
seven squared minus nine
times seven plus 14
so that's 49 minus 63
plus 14.
49 plus 14 is 63
minus 63 equals zero.
So x equals seven checks out.
And let's try x equals two.
Two squared minus nine
times two plus 14,
so that's four minus 18
plus 14,
so that's plus 14 is 18 again
minus 18 equals zero,
so x equals two checks out.
So these are our solutions.
This is x squared
so typically you'll have two solutions.
Notice, if it's just x
and the linear equations
we saw before
we only had one solution.
So that's one thing
to keep in mind.
Now we have 5x squared minus
9x equals two.
Our previous examples had zeros
so first we have to subtract
two from both sides.
We have 5x squared minus 9x
minus two equals zero
and now we have to factor.
We're going to have 5x
plus two.
Oops...that should be
a minus two actually.
So that would be–
let's do a quick check.
5x squared minus 10x
plus x minus two,
that's 5x squared minus 9x
minus two
so that part checked out
and then we're going to solve.
So we have 5x plus one
equals zero
and x minus two equals zero.
Subtract one from both sides
and we have 5x equals negative one.
So x equals negative 1/5th.
And in this one we're going
to add two to both sides.
So x equals two.
Check our answer.
I'll do x equals negative 1/5th.
You can do the x equals two.
So x equals negative 1/5.
We have five times negative 1/5 squared
minus nine times negative 1/5.
So we have negative 1/5 squared
is positive one over 25,
minus times a negative
is positive,
so plus nine over five.
Five over 1/25 is 1/5th
because 25 divided by five is five.
Plus nine over five equals
10 over five, which is two.
Two equals two.
So x equals negative 1/5th
as an answer.
And x equals two
will be an answer also.
You check that one.
This one looks
a little bit more complicated,
but let's start with what
we know how to do.
We know we need zero
on this side,
but first let's multiply that out.
That way we're not going
to miss anything.
So we have 2x squared
plus 16x plus 23 equals,
remember this is x plus four
times x plus four
so don't forget to distribute everything.
x squared plus 4x
plus 4x plus 16.
So combining like terms
on the right side,
we have 2x squared plus 16x
plus 23 equals x squared
plus 8x plus 16.
Now we want zero
on one side
and everything else on the other.
So we're going to subtract
x squared from both sides.
We're going to subtract 8x
from both sides.
Notice how I'm lining them up
with the like terms
and we're going to subtract 16
from both sides.
The left side will be 2x squared
minus x squared is x squared.
16x minus 8x is plus 8x.
23 minus 16 is plus 7 equals...
each of these cancel out
so we're left with zero.
Now we have to factor.
So that is x plus seven
times x plus one equals zero.
So x plus seven equals zero.
x plus one equals zero.
Subtract seven from both sides
and we have x equals negative seven.
Subtract one from both sides
and we have x equals negative one.
I'll check negative one.
You can check
x equals negative seven.
x equals negative one.
We're going to put on the left side,
so two times negative one squared
plus 16 times negative one plus 23.
Two times negative one squared is two
minus 16 plus 23
so that's 25 minus 16
would be nine.
Now we check the other side.
And we have negative one
plus four,
all of that is squared
so that's three squared, which is nine.
We checked both sides,
both sides are equal
so x equals negative one
is the solution.
So make sure–
and then you would check negative seven.
The square root property
is when we solve equations,
we need to do the same thing
to both sides.
This includes other operations
other than just addition, subtraction,
multiplication, division.
We can take the square root
of both sides.
We can take both sides
to an exponent.
We can take the logarithm
of both sides.
We can square both sides
if we need to.
So there's a whole bunch
of things that we can do.
This time around we're going
to take the square root
of both sides of the equation,
but we need to remember one thing.
If we have positive three squared,
we get nine.
If we have negative three squared,
we also get positive nine.
But if we take
the square root of nine,
that's only three.
So we have to be aware
that when we take
the square root of both sides,
we need to take plus or minus
the square root of nine
to get plus or minus three.
So that's what
we're going to do here.
So x squared equals nine,
if we take the square root
of both sides,
because the square root 
of x squared is x,
we want plus or minus three.
We need to make sure
when we take the square root
of both sides
we take plus or minus.
And then the square root 
of nine is three.
Checking our answers
we put three in for x.
In the original equation
three squared equals nine
and negative three squared
also equals nine.
So again you have to check
both of them.
When we take the square root
of both sides,
we need the x squared by itself
so we're going to divide
both sides by 36.
This is just like
whenever we solve anything
we're having the x chunk
by itself.
So we're dividing both sides by 36.
So x squared equals
25 over 36,
taking the square root of both sides.
We have x equals,
we have plus or minus,
the square root of 25 is five
and the square root of 36 is six.
And then you plug it
back in to check.
At this point I showed you
enough checks
so you can plug it
back in yourself and check 'em.
x squared equals negative 100.
Is there any number squared
that equals a negative number?
There's no real numbers
that do this
so this one is going to be
no solution.
x minus five quantity squared
equals 81.
This is still a chunk
with an x in it
that's squared equals 81,
so we're going to take
the square root of both sides.
x minus five squared equals...
plus or minus the square root of 81.
So x minus five equals 
plus or minus nine.
So now we can finish solving
but now we have to set up
two equations
because we can't add five
to both sides
with this plus or minus here.
So we have x minus five
equals nine.
Or x minus five
equals negative nine.
Add five to both sides
and we have x equals 14.
Add five to both sides
on this one.
And we get x equals negative four.
Again you can check
your answers in here.
And it checks quickly
because 14 minus five is nine
and nine squared is 81.
The same thing with negative four.
What happens though
when we have this, we can't,
so we'll take the square root
of both sides
and we get x minus five.
Plus or minus
the square root of 15.
We can't really do anything
with the square root of 15
because it's an irrational number,
but we can still add five to both sides.
We have x equals five
and then we're going to do
plus or minus the square root of 15.
You can leave it written like that
or you can write it as x equals five
plus the square root of 15
and five minus
the square root of 15.
We'll skip the completing
the square section
because not all...
never mind we'll do it.
If you didn't do it in your class,
talk to your instructor.
But recall from Section 2-7,
if we complete the square,
we take half of this number
and square it.
So half of six is three,
square it is nine.
And then if we were to factor that
we would have x squared...oops...
x squared plus 6x plus nine
is equal to x plus three squared.
So we can solve this
by completing the square.
Recall, what we want to do here,
is look at our x squared
minus 6x
and we're going to add 13
to both sides here.
So that we can complete
the square on this side.
So half of this number
is negative three,
squared is positive nine
so we're going to add
a plus nine here.
But whatever we do to one side
of the equation,
we have to do to the other.
So, if we add nine to this side,
we also have to add
nine to that side.
Now this side factors
into x minus three squared
and that's going to be equal
to 13 plus nine is 22.
So now just like we did previously
we take the square root
of both sides.
And we have x minus three
equals the plus or minus
the square root of 22.
We can't do anything
with the square root of 22
so we're just add three 
to both sides.
We have x equals three
plus or minus the square root of 22.
The quadratic formula
can be used to solve equations also.
The standard form
of the quadratic equation
is Ax squared plus Bx plus C.
And using this,
the quadratic formula then
is x equals negative b plus or minus
the square root of b squared
minus 4ac
all over 2a.
If you want to get a song
in your head
and help you remember this,
go to YouTube
and search quadratic formula.
Pop goes the weasel.
I will refrain from singing for you
because it's not nice.
So let's solve this equation
using the quadratic formula.
Because we have
three times five is 15.
There's no factors of 15
that add to nine
so we can't factor it.
If you want to solve equations
that you can factor
using the quadratic formula
that works fine.
But if you can't factor it,
you have to use the quadratic formula
or you could complete the square.
So recall the quadratic formula
was x equals negative b plus or minus
the square root of b squared
minus 4ac
all over 2a.
So we're going to have
negative nine.
We have x equals,
remember that x equals
negative nine plus or minus
the square root of b squared,
so nine squared minus four 
times a is three
and c is five
all over two times three.
So that's equal to negative nine
plus or minus
the square root of 81
minus...let's see...
12 times five is 60
all over six,
so that's negative nine
plus or minus
the square root of 21
over six.
We can't reduce that root anymore
so that's our answer.
You cannot simplify
the nine over six
because we can't simplify
the square root of 21.
If we're going to simplify,
we have to simplify
over the whole fraction.
The discriminant of a quadratic equation
is what's under the radical.
Remember that x equals
negative b plus or minus
the square root of
b squared minus 4ac
all over 2a.
This square root part tells us
what kind of solutions
we're going to have.
And sometimes that's all 
we need to know
is what type of solutions
are we going to have,
real solutions, are we going to have
one solution, two solutions.
So, if b squared minus 4ac
is bigger than zero,
think about that,
if this is bigger than zero
we have that plus or minus
the square root of a number.
If it equals zero,
there's going to be nothing here
so then we're not going to have
the plus or minus
so we would just have
negative b over 2a.
So that means that would
only be one solution.
So greater than zero
there's two real solutions.
Equal zero, there's one real solution.
If it's less than zero,
we would have a negative
under our square root
and that means 
there's no real solutions.
The solutions would only
be complex.
So this one we're just going
to determine
the type of solutions
we're going to have
and the number of solutions.
So we're just going to use
the b squared minus 4ac.
We're just going to use
that part.
So we have negative six
is our b squared
minus four times four
times three.
So we have 36 minus 16
times three is 48.
36 minus 48 is negative 12
so that means there's
no real solutions.
Because negative 12
is less than zero.
Math 94 students 
can skip number eight.
And then we're going to talk
about graphing.
Graphing a quadratic function.
Remember when we were graphing
absolute value equations,
we found the vertex,
and then we plotted the points
around the vertex.
We're going to do
the same thing
when we graph
a quadratic equation.
And then a couple other things
that we need to know.
The axis of symmetry.
We didn't really talk about that
with the absolute value,
but it's the same
with the absolute value.
The axis of symmetry
is the vertical line
that goes through the vertex.
So remember when we graphed
our absolute value equations,
if we subtract one
from both sides here,
we have y minus one
equals x plus five squared,
so our vertex is at–
this was the opposite,
our vertex is going
to be negative five,
and positive one.
Because it's the minus here.
So our vertex is at negative five,
positive one.
Our axis of symmetry it's going
to go through this x value
so it's going to be 
x equals negative five.
So if we plot negative five,
positive one, that's right here.
And then we have to find
some other values.
So we're going to start here.
We're going to use
x equals negative four
and negative three.
x equals negative six
and negative seven.
So x equals negative four.
We have negative four
plus five squared plus one
that would be one,
negative four plus five is one,
one squared is one
plus one is two
so our point is going to be
at negative four, two.
And we're going to do
x equals negative three also.
We have negative three
plus five squared plus one
so that's negative three plus seven
is positive two.
Positive two squared is four
plus one is five.
And that's something
that you can do
in your calculator,
if you need to.
So two of our points
are negative four, positive two,
and negative three, positive five.
And the same thing would happen
if we put in negative six.
This would be positive two
and negative seven
would be positive five.
Again those are ones
you should put in.
And then we graph.
And it looks like this.
The maximum or minimum value
of the function,
notice the function has
the smallest y value is at the vertex
because it's pointing down.
So it's a minimum value.
At y equals one.
That's the vertex value.
It's at y equals one.
If it was pointing up,
it would be a maximum value
at the vertex.
Another method to graphing
quadratic functions
is to find the vertex using the formula
x equals negative b over 2a.
Notice this when it looks like
the quadratic formula
only the square root part
is equal to zero.
Once you find the vertex,
you can find the intercepts.
And the intercepts work the same way
as when we found the intercepts
for a linear equation.
So let's go ahead and look at this.
So the vertex is at,
we have y equals x squared
plus 10x plus 27.
The vertex is at the x-value.
Equals negative b,
so negative 10 over two times,
there's nothing in front of the x
so it's two times one.
So the x-value is negative five.
To find the y-value of the vertex,
we're going to put
negative five in for x,
so y equals negative five squared
plus 10 times negative five
plus 27.
So that's 25 minus 50 plus 27,
so that would be positive two.
So our vertex–
I'm going to erase all the ink,
you can pause for a second
to write down the work.
Our vertex is at negative five,
positive two.
That means our axis of symmetry
is at x equals negative five.
Our x-intercepts are when
y equals zero.
So we're going to put...
We're going to put this in for y
and we have x squared
plus 10x plus 27.
That would be...
that doesn't factor
because we would solve
for x here.
The factors of 27
are three and nine.
That would be 12.
So that does not factor.
That's not good.
Let's make this a little bit different
so that it factors
so that we can show you
what happens a little bit better.
Let's see, what can we put in here
to make it factor.
We're going to change
this whole equation a little bit.
Let's see, if we made this 12.
Yeah, let's make this...
y equals.
y equals x squared plus 12x
plus 27.
So if we do that, our vertex,
so we're starting over completely.
I'm very sorry about that.
Our vertex is at x equals
negative 12
over two times one
which would be negative six.
And then our y-value
of the vertex
is going to be negative six squared
plus 12 times negative six plus 27.
So that's 36 minus 72
plus 27.
So that would be 63 minus 72
is negative nine.
So our vertex is at negative six,
negative nine.
That means our axis of symmetry
is at x equals negative six.
Let's erase some of this work
so we have other places,
some places to do work
for the other stuff.
So then our x-intercepts
is when y equals zero.
So we're going to do
zero equals x squared
plus 12x plus 27.
So zero equals x plus nine
times x plus three
so x plus nine equals zero
because we're solving for x.
That's why I factored.
Just like we did earlier.
And x plus three equals zero.
So x equals negative nine
and x equals negative three
by subtracting nine
and subtracting three.
So our x-intercepts are
at negative nine, zero
and negative three, zero.
Again I'm going to erase
so I have some room to work.
All right, and then we just need
our y-intercept.
Our y-intercept is when
x equals zero.
So now we're going
to put zero in for x.
So we have y equals zero squared
plus 12 times zero plus 27
so y equals zero plus zero
plus 27
so y equals 27.
So our y-intercept is at zero, 27.
That one is going to be
a little harder to graph,
but let's see what happens.
These are all multiples of three,
x and y,
so let's let our increment
be three.
So this will be
three and three
so we go three, six, nine, 12.
So negative six, negative nine,
so we have negative three,
negative six
and then down negative three,
negative six, negative nine is here.
Our axis of symmetry is
at x equals negative six.
So I'll draw a vertical line there.
Our x-intercepts are
at negative nine, zero.
So this was negative three,
negative six, negative nine is here.
And negative three, zero,
negative three, zero is there.
And our y-intercept is at 27
so three, six, nine, 12, 15, 18,
21, 24, 27.
And our axis of symmetry says,
if this is symmetrical
it's going to flip over
this axis of symmetry.
So when we do that,
we have zero, 27 here.
That's two away from the vertex.
So two away from the vertex
on the other side,
we're going to have
the 27 value also.
And then we can connect our points.
And that's what our graph looks like.
And it has a minimum value again
because it's pointed up.
At y equals negative nine.
That's it for today.
