Intermediate Algebra (11th edition)

OBJECTIVE 3 Solve a formula for a specified variable, where factoring is

necessary.In Section 2.2we solved certain formulas for variables. In some cases,

factoring is required to accomplish this.

A rectangular solid has the shape of a box, but is solid. See FIGURE 2. The surface

area of any solid three-dimensional figure is the total area of its surface. For a rectan-

gular solid, the surface area is

a= 2 HW+ 2 LW+ 2 LH.

a

SECTION 6.5 Solving Equations by Factoring 349

NOW TRY
EXERCISE 9
Solve the formula for H.

a= 2 HW+ 2 LW+ 2 LH

NOW TRY ANSWER

9. H=a 2 W-+^2 LW 2 L

NOW TRY

CAUTION InExample 9,we must write the expression so that the specified

variable is a factor. Then we can divide by its coefficient in the final step.

In Section 5.3,we saw that the graph of is a parabola. In general, the graph of

is a parabola, and the x-intercepts of its graph give the real number solutions of the

equation

A graphing calculator can locate these x-intercepts (called zerosof the function)

for. Notice that this quadratic expression was found on the left

side of the equation in Example 2(a)earlier in this section, where the equation was

written in standard form. The x-intercepts (zeros) given with the graphs in FIGURE 3

are the same as the solutions found in Example 2(a).

ƒ 1 x 2 = 2 x^2 + 3 x- 2

ax^2 + bx+ c= 0.

ƒ 1 x 2 = ax^2 +bx+ c, aZ0,

ƒ 1 x 2 =x^2

CONNECTIONS

–4

–6 6

4 4

–4

–6 6

FIGURE 3

L

H

W

Rectangular solid

a = 2HW + 2 LW + 2 LH

FIGURE 2

Using Factoring to Solve for a Specified Variable

Solve the formula for L.

To solve for the length L, treat Las the only variable and treat all other variables

as constants.

Subtract 2HW.

Factor out L.

Divide by 2W+ 2 H.

a- 2 HW

2 W+ 2 H

= L, or L=

a- 2 HW

2 W+ 2 H

a- 2 HW= L 12 W+ 2 H 2

a- 2 HW= 2 LW+ 2 LH

a= 2 HW+ 2 LW+ 2 LH

a= 2 HW+ 2 LW+ 2 LH

EXAMPLE 9

H, W, and Lrepresent height,

width, and length.

For Discussion or Writing

Solve each quadratic equation using the zero-factor property. Then support the

solution(s) with a graphing calculator.

1. x^2 - 6 x- 7 = 0 2. x^2 - 6 x+ 9 = 0 3. x^2 = 4

We must isolate

the L-terms.

This is a

key step.