Solving Polynomial Equations by

Graphing

Types of Equations

Quadratic - has the form ax2 + 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 ax3 + bx2 + 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 ax4 + bx3 + cx2 + dx + e = 0

Highest exponent is four (this is the degree)

The most real solutions it has is four.

Types of EquationsThese 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 EquationsWhen 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 EquationsThe 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 EquationsSolve: x2 - 4 = yReplace y with 0.Plug in x2 - 4 into your calculator. Graph it and let’s look at the graph.

Solving EquationsWhen 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 x-

intercepts.

Solving EquationsSo 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 EquationsThis 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 EquationsFind the solutions to

f(x) = x2 - 5x + 6.

x = 3, 2Find the zero’s of

0 = x2 - 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 EquationsLet’s look at some cubic equations.

x3 - 1 = 0

x = 1

x3 - 6x + 1 = f(x)

Has three solutions.

Solving EquationsWhen we are solving cubic

equations, we will have either 3, 2, or 1 real solution. You should never have no solutions.

Solving EquationsWhat about quartic

equations?They look like W or M.They could have four, three,

two, one, or no solution.

Solving EquationsLet’s look at your graphing

equations worksheet.

FactoringFor 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

x2 - 4 = 0Let’s try the first graphing example

and factor it.to factor x2 - 4 we use difference of squares.

x2 - 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) = 0So x - 2 = 0, meaning x = 2or x + 2 = 0, meaning x = -2So our solutions are x = 2 and x = -

2.

Factoring Quadratics

Find the roots by factoring: 2x2 + 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 = 0x = -6x - 2 = 0x = 2So x = -6 or x = 2.

Quadratic FormulaThe 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

ax2 + bx + c = 0a 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 b2 4ac

2a

Quadratic Formula

Let’s look at an example.3x2 - 4x + 3 = 0a = ?b = ?c = ?

a = 3b = -4c = 3

Quadratic Formula

Now let’s plug it in.b = -4, so -b = -(-4) = 4

4 ( 4)2 4(3)(3)

2(3)

Quadratic Formula

Simplify

4 16 36

6

Quadratic Formula

Keep going now.

4 20 4 2 5

6 6

i

Quadratic Formula

4 2 5

6

4 - 2 5

6

ix

ix

Quadratic Formula

Find the zeros ofr2 - 7r -18 = 0

Quadratic Formula

Find the zeros of r2 - 7r -18 = 0

a = 1

b = -7

c = -18

Quadratic Formula

Find the zeros ofr2 - 7r -18 = 0

27 ( 7) 4(1)( 18)

2(1)r

Quadratic Formula

Simplify7 49 ( 72)

2

7 121 7 11

2 2

r

Quadratic Formula

Now let’s examine our solution.

We can break this into two equations.

7 11

2r

Quadratic Formula

Now we can get our two solutions.

7 11 189

2 27 11 4

22 2

r

r

Quadratic Formula

Now you try some.pg. 35710 - 13 (only solve them using the quadratic formula)