__Definition: __The __zero- product property __ says
that if a and b are numbers and if ab = 0, then

a = 0 or b = 0 (or both).

__Definition:__ A __quadratic equation__ is an equation that can be
written ax^{2} + bx + c = 0,

where a, b, and c are numbers and a ≠ 0.

** Solving Quadratic Equations**

To solve a quadratic equation, we must find all possible values for x that make
ax^{2} + bx +

c = 0. Factoring is usually a helpful way to solve quadratic equations. To use
factoring,

move all nonzero terms to one side of the equal sign so that the other side is
zero. Then

use the zero-product property.

**Examples:**

1. Solve the equation x^{2} – 5x – 24 = 0.

Only the left-hand side (LHS) of the equation has non zero terms , so no movement

of terms is necessary. Factor the LHS and use the zero-product property (ZPP):

x2 – 5x – 24 = 0

Factor: (x + 3)(x – 8) = 0

ZPP: x + 3 = 0 or x – 8 = 0

Solve for x: x = –3 or x = 8

2. Solve 2x^{2} + 18x – 72 = 0 for x.

All nonzero terms are on the LHS, so no movement of terms is needed.

2x^{2} + 18x – 72

Factor: 2(x + 12)(x – 3) = 0

ZPP: x + 12 = 0 or x – 3 = 0 (Note: 2 ≠ 0, so we don’t write it)

Solve for x: x = –12 or x = 3

3. Solve the equation 6x^{2} – 27x = –12.

This equation has nonzero terms on both sides of the equal sign. To make it

possible to solve, move the –12 to the LHS by adding 12 to both sides:

6x^{2} – 27x + 12 = 0

Factor: 3(2x – 1)(x – 4) = 0

ZPP: 2x – 1 = 0 or x – 4 = 0 (Note: 3 ≠ 0)

Solve for x: x = ½ or x = 4

4. Solve 16x^{2} = 1.

Subtract 1 from both sides: 16x^{2} – 1 = 0

Factor: (4x – 1)(4x + 1) = 0 (Note: This is a ‘special’ polynomial.)

ZPP: 4x – 1 = 0 or 4x + 1 = 0

Solve for x: x = ¼ or x = -¼

5. Solve: –12x^{2} = –17x + 6

This quadratic equation has nonzero terms on both sides of the equal sign, so we

must move all the terms to one side. Add 17x – 6 to both sides and solve.

–12x^{2} + 17x – 6 = 0

Factor: –1(4x – 3)(3x – 2) = 0 (Note: –1 ≠ 0)

ZPP: 4x – 3 = 0 or 3x – 2 = 0

Solve for x: x = ¾ or x =

6. Solve for x: x(x – 2) = –1

Make one side equal zero: x(x – 2) + 1 = 0

Distribute : x2 – 2x + 1 = 0

Factor: (x – 1)^{2} = 0 (Note: This is a ‘special’ polynomial.)

ZPP: x – 1 = 0 (Note: No need to write x – 1 = 0 twice.)

Solve for x: x = 1

**Solving Other Polynomial Equations **

Solving other polynomial equations is done just like the quadratic equations:
Set one side

of the equation to zero, factor, use the zero-product property, and solve for x.

**Examples:**

1. Solve 4x^{3} + 16x^{2} + 15x = 0.

This polynomial is already equal to zero, so all we need do is factor, use the
ZPP,

and solve for x.

4x^{3} + 16x^{2} + 15x = 0

Factor: x(2x + 5)(2x + 3) = 0

ZPP: x = 0 or 2x + 5 = 0 or 2x + 3 = 0

Solve for x: x = 0 or x = -5/2 or x = -3/2

2. Solve x3 = 2x^{2} + 99.

Get LHS equal to zero: x3 – 2x^{2} – 99 = 0

Factor: x(x – 11)(x + 9) = 0

ZPP: x = 0 or x – 11 = 0 or x + 9 = 0

Solve for x: x = 0 or x = 11 or x = –9

3. Solve 7x^{3} – 14x^{2} = 0.

Set one side equal to zero: Already done.

Factor: 7x^{2}(x – 2) = 0

ZPP: 7x^{2} = 0 or x – 2 = 0

Solve for x: x = 0 or x = 2

4. Solve x^{3} + 18 = 2x^{2} + 9x by factoring.

Set one side equal to zero: x^{3} + 18 – 2x^{2} – 9x = 0

Rewrite: x^{3} – 2x^{2} – 9x + 18 = 0

Factor: (xa^{2} – 9)(x – 2) = 0

(x + 3)(x – 3)(x – 2) = 0

ZPP: x + 3 = 0 or x – 3 = 0 or x – 2 = 0

Solve for x: x = –3 or x = 3 or x = 2

5. Solve 30x^{3} – 3x^{2} – 9x = 0.

Set one side equal to zero: Already done.

Factor: 3x(5x – 3)(2x + 1) = 0

ZPP: 3x = 0 or 5x – 3 = 0 or 2x + 1 = 0

Solve for x: x = 0 or x = 3/5 or x = -½

6. Solve 2(x – 1)^{2} = 2x(x^{2} – 20) + (8x^{2} + 2).

Set one side equal to zero: 2(x – 1)^{2} – 2x(x^{2} – 20) – (8x^{2} + 2) = 0

Simplify: 2(x^{2} – 2x + 1) – 2x(x^{2} – 20) – (8x^{2} + 2) = 0

2x^{2} – 4x + 2 – 2x^{3} + 40x – 8x^{2} – 2 = 0

Rewrite: –2x^{3} – 6x^{2} + 36x = 0

Factor: –2x(x + 6)(x – 3) = 0

ZPP: –2x = 0 or x + 6 = 0 or x – 3 = 0

Solve for x: x = 0 or x = –6 or x = 3

**Finding x-Intercepts**

Finding the x- intercepts of any polynomial is like finding the x- intercepts of a
line : set

the equation equal to zero and solve for x. To find the y-intercept, plug zero
into the

given formula everywhere an __x__ appears and simplify.

**Examples:**

1. Find the x- and y-intercepts of the parabola f (x) = 3x^{2} – 9x + 54

To find the y-intercept, find f(0): (0)^{2} – 9(0) + 54 = 54

To find the x-intercept(s), solve 3x^{2} – 9x + 54 = 0:

Factor: 3(x + 3)(x – 6) = 0

ZPP: x + 3 = 0 or x – 2 = 0 (Note: 3 ≠ 0)

Solve for x: x = –3 or x = 2

Intercepts: (0, 54), (–3, 0), and (6, 0)

The graph of this function looks like the following. Note
the x- and y-intercepts

match what was calculated .

2. Find the intercepts of the graph of the polynomial P(x)
= x^{3} + 7x^{2} + 10x.

To find the y-intercept, find P(0): (0)^{3} + 7(0)^{2} + 10(0) = 0.

To find the x-intercept(s), set P(x) = 0 and solve for x:

x^{3} + 7x^{2} + 10x = 0

Factor: x(x + 2)(x + 5) = 0

ZPP: x = 0 or x + 2 = 0 or x + 5 = 0

Solve for x: x = 0 or x = –2 or x = –5

Intercepts: (0, 0), (–2, 0), and (–5, 0)

The graph of this function looks like the following. Note
the x- and y-intercepts

match what was calculated.

3. Find the x- and y-intercepts of the function f(x) = x^{3}
– 3x^{2} – 9x + 27.

For the y-intercept, find f(0): (0)^{3} – 3(0)^{2} – 9(0) + 27 = 27.

For the x-intercept(s), set f(x) = 0 and solve for x:

x^{3} – 3x^{2} – 9x + 27 = 0

Factor: (x + 3)(x – 3)^{2} = 0

ZPP: x + 3 = 0 or x – 3 = 0

Solve for x: x = –3 or x = 3

Intercepts: (0, 27), (–3, 0), and (3, 0)

The graph of this function looks like the following. Note
the x- and y-intercepts

match what was calculated.

Solve the fol lowing equations by factoring.

1. 3x^{2} – 5x + 2 = 0

2. x^{3} – x = 0

3. x^{4} + 6x^{3} + 8x^{2} = 0

4. 3x^{2} + 2x = 8

5. 8x^{2} + 10x = 3

6. 3x^{2} = 2x + 8

7. –42x^{3} – 148x^{2} – 96x = 0

8. 8x^{4} = 32x^{2}

9. 4x^{2} + 40 = 44x

Find the x- and y-intercepts of the following polynomial
functions.

10. f(x) = 7x^{3} + 21x^{2} – 7x – 21

11. f(x) = 6x^{2} + 7x – 20

12. f(x) = 48x^{3} + 212x^{2} + 224x

13. P(x) = –4x^{2} + 14x^{2} – 12x

14. P(x) = 7(x – 2)^{2} – 28

15. P(x) = (x – 3)(x – 2)(x + 4)