Beginning Algebra, 11th Edition

392 CHAPTER 6 Factoring and Applications

OBJECTIVES Galileo Galilei developed theories to explain physical phenomena and set up experi-

ments to test his ideas. According to legend, Galileo dropped objects of different

weights from the Leaning Tower of Pisa to disprove the belief that heavier objects fall

faster than lighter objects. He developed the formula

describing the motion of freely falling objects. In this formula, dis the distance in

feet that an object falls (disregarding air resistance) in tseconds, regardless of weight.

The equation is a quadratic equation. A quadratic equation contains a

second-degree term and no terms of greater degree.

d= 16 t^2

d= 16 t^2

Solving Quadratic Equations by Factoring

6.5

1 Solve quadratic

equations by

factoring.

2 Solve other

equations by

factoring.

Galileo Galilei (1564–1642)

Quadratic Equation

A quadratic equationis an equation that can be written in the form

where a, b, and care real numbers, with aZ0.

ax^2 bxc0,

The form is the standard formof a quadratic equation.

Quadratic equations

Of these quadratic equations, only is in standard form.

We have factored many quadratic expressionsof the form In this

section, we use factored quadratic expressions to solve quadratic equations.

ax^2 + bx+c.

x^2 + 5 x+ 6 = 0

x^2 + 5 x+ 6 = 0, 2 x^2 - 5 x= 3, x^2 = 4

ax^2 + bx+ c= 0

OBJECTIVE 1 Solve quadratic equations by factoring. We use the zero-factor

propertyto solve a quadratic equation by factoring.

Zero-Factor Property

Ifaandbare real numbers and if then or

That is, if the product of two numbers is 0, then at least one of the numbers must

be 0. One number mustbe 0, but both maybe 0.

ab0, a 0 b0.

Using the Zero-Factor Property

Solve each equation.

(a)

The product is equal to 0. By the zero-factor property, the only

way that the product of these two factors can be 0 is if at least one of the factors

equals 0. Therefore, either or

Zero-factor property

Solve each equation.

x= Divide each side by 2.

1

2

x=-3 2x= 1

x+ 3 = 0 or 2 x- 1 = 0

x+ 3 = 0 2 x- 1 = 0.

1 x+ 3212 x- 12

1 x+ 3212 x- 12 = 0

EXAMPLE 1

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