Beginning Algebra, 11th Edition

Solving Quadratic Equations with Double Solutions

Solve each equation.

(a)

Factor the perfect square trinomial.

Zero-factor property

Because the two factors are identical, they both lead to the same solution. (This is

called a double solution.)

Add 11.

CHECK

Let.

✓ True

The solution set is.

(b)

Standard form

Factor the perfect square trinomial.

Zero-factor property

Solve the equation.

is a double solution.

Checkby substituting in the original equation. The solution set is

NOW TRY

E

5

3 F.

5

3

5

t= 3

5

3

3 t= 5

3 t- 5 = 0 or 3 t- 5 = 0

13 t- 522 = 0

9 t^2 - 30 t+ 25 = 0

9 t^2 - 30 t=- 25

5116

0 = 0

121 - 242 + 121  0

112 - 221112 + 121  0 z= 11

z^2 - 22 z+ 121 = 0

z= 11

z- 11 = 0 or z- 11 = 0

1 z- 1121 z - 112 = 0 a^2 =a#a

1 z- 1122 = 0

z^2 - 22 z+ 121 = 0

EXAMPLE 5

396 CHAPTER 6 Factoring and Applications

CAUTION Each of the equations in Example 5has only onedistinct solution.

There is no need to write the same number more than once in a solution set.

OBJECTIVE 2 Solve other equations by factoring. We can also use the zero-

factor property to solve equations that involve more than two factors with variables.

(These equations are notquadratic equations. Why not?)

Solving Equations with More Than Two Variable Factors

Solve each equation.

(a)

Factor out 6z.

Factor

By an extension of the zero-factor property, this product can equal 0 only if at least

one of the factors is 0. Write and solve three equations, one for each factor with a

variable.

6 z 1 z+ 121 z- 12 = 0 z^2 - 1.

6 z 1 z^2 - 12 = 0

6 z^3 - 6 z = 0

EXAMPLE 6

NOW TRY
EXERCISE 5
Solve.

4 x^2 - 4 x+ 1 = 0

NOW TRY ANSWER

5. E^12 F

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