Intermediate Algebra (11th edition)

518 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

Solving an Equation That Is Quadratic in Form by Substitution

Step 1 Define a temporary variable u,based on the relationship between

the variable expressions in the given equation. Substitute uin the orig-

inal equation and rewrite the equation in the form

Step 2 Solve the quadratic equation obtained in Step 1by factoring or

the quadratic formula.

Step 3 Replace uwith the expression it defined in Step 1.

Step 4 Solve the resulting equations for the original variable.

Step 5 Checkall solutions by substituting them in the original equation.

au^2 +bu+c= 0.

Solving Equations That Are Quadratic in Form

Solve each equation.

(a)

Step 1 Because of the repeated quantity substitute ufor (See

Example 5(b).)

Let

Step 2 Factor.

or Zero-factor property

or Solve for u.

Step 3 or Substitute 4x 3 for u.

Step 4 or Solve for x.

or

Step 5 Check that the solution set of the original equation is

(b)

Substitute ufor. (See Example 5(c).)

Let

Factor.

or Zero-factor property

or Solve for u.

or

or Cube each side.

or

Check that the solution set is E^278 , 64F. NOW TRY

x= x= 64

27

8

1 x1/3 23 = a 1 x1/3 23 = 43

3

2

b

3

x 1/3 = x1/3= 4 u=x1/3

3

2

u= u= 4

3

2

2 u- 3 = 0 u - 4 = 0

12 u- 321 u- 42 = 0

2 u^2 - 11 u+ 12 = 0 x1/3=u; x2/3=u^2.

x1/3

2 x2/3- 11 x1/3 + 12 = 0

E

1

8 ,

1

2 F.

x=

1

2

x=

1

8

4 x= 4 x= 2

1

2

4 x- 3 =- 4 x- 3 =- 1 -

5

2

u=- u=- 1

5

2

2 u+ 5 = 0 u + 1 = 0

12 u+ 521 u+ 12 = 0

2 u^2 + 7 u+ 5 = 0 u= 4 x-3.

214 x- 322 + 714 x- 32 + 5 = 0

4 x- 3, 4 x-3.

214 x- 322 + 714 x- 32 + 5 = 0

EXAMPLE 7

Don’t stop here.

NOW TRY
EXERCISE 7
Solve each equation.

(a)

(b) 2 x2/3- 7 x1/3+ 3 = 0

- 10 = 0

61 x- 422 + 111 x- 42

NOW TRY ANSWERS

7. (a)E^32 ,^143 F (b)E^18 , 27F