Solving Polynomial
Equations by Graphing
Types of Equations
Quadratic
- has the form
2
ax + bx + c = 0
Highest exponent is two
(this is the degree)
The most real solutions it
has is two.
Types of Equations
Cubic
- has the form
3
2
ax + bx + cx + d = 0
Highest exponent is three
(this is the degree)
The most real solutions it
has is three.
Types of Equations
Quartic
- has the form
4
3
2
ax + bx + cx + dx + e = 0
Highest exponent is four
(this is the degree)
The most real solutions it
has is four.
Types of Equations
These
keep on going up as
the highest exponent
increases.
You don’t need to know the
names above quartic, but
you do need to be able to
give the degree.
Solving Equations
When
we talk about solving
these equations, we want to
find the value of x when
y = 0.
Instead of ‘solve’ we call
this finding ‘zeros’ or
‘roots’.
Solving Equations
Get
all the x or constant
terms on one side.
If
you have a y or f(x),
replace it with 0.
Solving Equations
The
first way we are going
to solve these equations is
by graphing. (Yeah!!! More
calculator stuff!!)
Go to the graph menu on
your calculator.
Solving Equations
Solve:
2
x
-4=y
Replace y with 0.
2
Plug in x - 4 into your
calculator.
 Graph it and let’s look at
the graph.
Solving Equations
When
we talk about the
graph and we are looking for
places where y = 0, where
will these points be?
On the x-axis.
So we are looking for the xintercepts.
Solving Equations
So
where does this graph
cross the x-axis?
(2, 0) and (-2, 0)
If you can’t tell from looking
at the graph, go to F5 (gsolv)
and then F1 (root).
Solving Equations
This
should give you the first
zero, to get to the second, hit
the right arrow button.
Note: the zeros should be on
the screen. If you can’t see
the x-intercepts, make your
window bigger.
Solving Equations
So
the solutions to this
equation are x = 2 or x = -2.
Solving Equations
Find
the solutions to
2
f(x) = x - 5x + 6.
x = 3, 2
Find the zero’s of
2
0 = x - 4x + 4
x=2
Solving Equations
How
do we check our
solutions?
Plug in and see if the
equation simplifies to 0.
Solving Equations
Let’s
look at quadratic
equations for a minute.
How many solutions should
you look for?
Two, one or zero.
Solving Equations
Let’s
look at some cubic
equations.
x3 - 1 = 0
x=1
3
x - 6x + 1 = f(x)
Has three solutions.
Solving Equations
When
we are solving cubic
equations, we will have either
3, 2, or 1 real solution. You
should never have no
solutions.
Solving Equations
What
about quartic
equations?
They look like W or M.
They could have four, three,
two, one, or no solution.
Solving Equations
Let’s
look at your graphing
equations worksheet.
Factoring
For
these last two methods
for solving equations, we will
be looking at only quadratic
equations (degree 2).
The next method we will look
at is factoring.
Factoring Quadratics
We
know that quadratic
equations are set equal to 0.
We will factor the trinomial
and set each factor equal to 0
to find our solutions.
Factoring Quadratics
2
x
-4=0
Let’s try the first graphing
example and factor it.
2
to factor x - 4 we use
difference of squares.
2
x - 4 = (x - 2)(x + 2) = 0
Factoring Quadratics
Okay,
let’s take a side note
for a second.
If we multiply two numbers
and get a product of 0, what
do the factors have to be?
3x = 0, what does x have to
be?
Factoring Quadratics
if
ab = 0, what do we know
about a or b.
Either a has to be 0, b has to
be 0, or they both can be
zero.
This is the only way to get a
product of 0.
Factoring Quadratics
Okay,
back to factoring.
(x - 2)(x + 2) = 0
So x - 2 = 0, meaning x = 2
or x + 2 = 0, meaning x = -2
So our solutions are x = 2
and x = -2.
Factoring Quadratics
Find
the roots by factoring:
2
2x + 8x - 24 = 0
First, factor 2x2 + 8x - 24.
2(x + 6)(x - 2).
Set each factor (that contains
an x) equal to zero.
Factoring Quadratics
x
+6=0
x = -6
x - 2 = 0
x = 2
So x = -6 or x = 2.
Quadratic Formula
The
last method we will use
to solve quadratic equations
is the quadratic formula.
This is the only method that
will ALWAYS work when
trying to solve a quadratic
equation.
Quadratic Formula
All the quadratic formula is
is plugging in numbers.
You don’t need to worry
about memorizing it. They
give it to you on the SOL

Quadratic Formula
Let’s look back the the
general form of a quadratic
equation.

Quadratic Formula
2
 ax
+ bx + c = 0
 a is the coefficient of the squared
term.
 b is the coefficient of the x term.
 c is the constant.
Quadratic Formula
If
one of these three terms
doesn’t exist, then the
coefficient of that term will
be ____?
0
Quadratic Formula
what
is the quadratic
formula?

b  b  4ac
2a
2
Quadratic Formula
Let’s
look at an example.
2
3x - 4x + 3 = 0
a = 3
a = ?
b = -4
b = ?
c = 3
c = ?
Quadratic Formula
Now
let’s plug it in.
b = -4, so -b = -(-4) = 4
4  (4)  4(3)(3)
2(3)
2
Quadratic Formula
Simplify
4  16  36
6
Quadratic Formula
Keep going now.
4  20 4  2i 5

6
6
Quadratic Formula
4  2i 5
x
6
4 - 2i 5
x
6
Quadratic Formula
Find the zeros of
2
r - 7r -18 = 0
Quadratic Formula
Find the zeros of
2
r - 7r -18 = 0
a=1
b = -7
c = -18
Quadratic Formula
Find the zeros of
2
r - 7r -18 = 0
7  (7)  4(1)(18)
r
2(1)
2
Quadratic Formula
Simplify
7  49  (72)
r

2
7  121 7  11

2
2
Quadratic Formula
Now let’s examine our
solution.
7  11
r
2
We can break this into two
equations.
Quadratic Formula
Now we can get our two
solutions.
7  11 18
r
 9
2
2
7  11 4
r

 2
2
2
Quadratic Formula
Now
you try some.
pg. 357
10 - 13 (only solve them
using the quadratic formula)