ADVERTISEMENT

Solving Quadratic Equations

6.9

by Factoring

OBJECTIVES

6.9

1.

Solve a quadratic equation by factoring

2.

Find the zeros of a quadratic function

The factoring techniques you have learned provide us with tools for solving equations that

can be written in the form

This is a quadratic equation in one variable, here x.

You can recognize such a quadratic equation by

2

ax

bx

c

0

a

0

the fact that the highest power of the variable x is

the second power.

in which a, b, and c are constants.

2

An equation written in the form ax

bx

c

0 is called a quadratic equation in

standard form. Using factoring to solve quadratic equations requires the zero-product

rule, which says that if the product of two factors is 0, then one or both of the factors must

be equal to 0. In symbols:

Rules and Properties: Zero-Product Rule

If a b

0, then a

0 or b

0 or a

b

0.

Let’s see how the rule is applied to solving quadratic equations.

Example 1

Solving Equations by Factoring

Solve.

2

x

3x

18

0

Factoring on the left, we have

NOTE

(x

6)(x

3)

0

To use the zero-product

rule, 0 must be on one side of

the equation.

By the zero-product rule, we know that one or both of the factors must be zero. We can then

write

x

6

0

or

x

3

0

Solving each equation gives

NOTE

Graph the function

2

y

x

3x

18

x

6

or

x

3

on your graphing calculator.

The solutions to the equation

The two solutions are 6 and

3 and the solution set is written as x x

3, 6 or simply

2

0

x

3x

18 will be the x

3, 6 .

coordinates of the points on

The solutions are sometimes called the zeros, or roots, of the equation. They represent

the curve at which y

0. Those

2

the x coordinates of the points where the graph of the equation y

x

3x

18 crosses

are the points at which the

the x axis. In this case, the graph crosses the x axis at ( 3, 0) and (6, 0).

graph intercepts the x axis.

463

ADVERTISEMENT

0 votes

Parent category: Education