Example 1 Equation with Rational Roots Solve x 2 + 14x + 49 = 64 by using the Square Root Property....

Equation with Rational Roots

Solve x

2 + 14x + 49 = 64 by using the Square Root Property.

Original equation

Factor the perfect square trinomial.

Square Root Property

Subtract 7 from each side.

Equation with Rational Roots

x = 1 x = –15 Solve each equation.

Answer: The solution set is {–15, 1}.

x = –7 + 8 or x = –7 – 8 Write as two equations.

Check: Substitute both values into the original equation.

x

2 + 14x + 49 = 64 x

2 + 14x + 49 = 64??

1

2 + 14(1) + 49 = 64 (–15)

2 + 14(–15) + 49 = 64??

1 + 14 + 49 = 64 225 + (–210) + 49 = 64

64 = 64 64 = 64

Equation with Irrational Roots

Solve x

2 – 4x + 4 = 13 by using the Square Root Property.

Square Root Property

Original equation

Factor the perfect square trinomial.

Add 2 to each side.

Write as two equations.

Use a calculator.

Equation with Irrational Roots

x

2 – 4x + 4 = 13 Original equation

x

2 – 4x – 9 = 0 Subtract 13 from each side.

y = x

2 – 4x – 9 Related quadratic function

Answer: The exact solutions of this equation are The approximate solutions are 5.61 and –1.61. Check these results by finding and graphing the related quadratic function.

Equation with Irrational Roots

Check Use the ZERO function of a graphing calculator. The approximate zeros of the related function are –1.61 and 5.61.

A. A

B. B

C. C

D. D

Solve x

2 – 4x + 4 = 8 by using the Square Root Property.

A.

B.

C.

D.

Complete the Square

Find the value of c that makes x

2 + 12x + c a perfect square. Then write the trinomial as a perfect square.

Step 1 Find one half of 12.

Answer: The trinomial x2 + 12x + 36 can be written as (x + 6)2.

Step 2 Square the result of Step 1. 62 = 36

Step 3 Add the result of Step 2 to x

2 + 12x + 36x

2 + 12x.

A. A

B. B

C. C

D. D

A. 9; (x + 3)2

B. 36; (x + 6)2

C. 9; (x – 3)2

D. 36; (x – 6)2

Find the value of c that makes x2 + 6x + c a perfect square. Then write the trinomial as a perfect square.

Solve an Equation by Completing the Square

Solve x2 + 4x – 12 = 0 by completing the square.

x2 + 4x – 12 = 0 Notice that x2 + 4x – 12 is not a perfect square.

x2 + 4x = 12Rewrite so

the left side is of the form x2 + bx.

x2 + 4x + 4 = 12 + 4

add 4 to

each side. (x + 2)2 = 16Write the

left side as a perfect square by factoring.

Solve an Equation by Completing the Square

x + 2 = ± 4 Square Root Property

Answer: The solution set is {–6, 2}.

x = – 2 ± 4Subtract 2

from each side.

x = –2 + 4 or x = –2 – 4 Write as two equations.

x = 2 x = –6 Solve each equation.

Equation with a ≠ 1

Solve 3x2 – 2x – 1 = 0 by completing the square.

3x2 – 2x – 1 = 0 Notice that 3x2 – 2x – 1 is not a perfect square.Divide by the coefficient of the quadratic term, 3.

Add to each side.

Equation with a ≠ 1

Write the left side as a perfect square by factoring. Simplify the right side.

Square Root Property

Equation with a ≠ 1

Answer:

x = 1 Solve each equation.

or Write as two equations.

A. A

B. B

C. C

D. D

Solve 2x2 + 11x + 15 = 0 by completing the square.

A.

B.

C.

D.

Equation with Imaginary Solutions

Solve x

2 + 4x + 11 = 0 by completing the square.

Notice that x

2 + 4x + 11 is not a perfect square.

Rewrite so the left side is of the form x

2 + bx.

Since , add 4 to each side.

Write the left side as a perfect square.

Square Root Property

Equation with Imaginary Solutions

Subtract 2 from each side.