1. Solve x 2 + 14x + 49 = 64 by using the Square Root Property.
Equation with Rational Roots
Solve x x + 49 = 64 by using the Square Root Property.
Original equation
Factor the perfect square trinomial.
Square Root Property
Subtract 7 from each side.
Example 1

2. x = –7 + 8 or x = –7 – 8 Write as two equations.
Equation with Rational Roots
x = –7 + 8 or x = –7 – 8 Write as two equations.
x = 1 x = –15 Solve each equation.
Answer: The solution set is {–15, 1}.
Check: Substitute both values into the original equation.
x x + 49 = 64 x x + 49 = 64 ?
(1) + 49 = 64 (–15) (–15) + 49 = 64 ?
= (–210) + 49 = 64
64 = = 64
Example 1

3. Solve x 2 – 4x + 4 = 13 by using the Square Root Property.
Equation with Irrational Roots
Solve x 2 – 4x + 4 = 13 by using the Square Root Property.
Original equation
Factor the perfect square trinomial.
Square Root Property
Add 2 to each side.
Write as two equations.
Use a calculator.
Example 2

4. x 2 – 4x + 4 = 13 Original equation
Equation with Irrational Roots
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.
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
Example 2

5. 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.
Example 2

6. A B C D Solve x 2 – 4x + 4 = 8 by using the Square Root Property. A.
Example 2

7. Concept

8. Step 2 Square the result of Step 1. 62 = 36
Complete the Square
Find the value of c that makes x x + c a perfect square. Then write the trinomial as a perfect square.
Step 1 Find one half of 12.
Step 2 Square the result of Step = 36
Step 3 Add the result of Step 2 to x x + 36 x x.
Answer: The trinomial x2 + 12x + 36 can be written as (x + 6)2.
Example 3

9. Find the value of c that makes x2 + 6x + c a perfect square
Find the value of c that makes x2 + 6x + c a perfect square. Then write the trinomial as a perfect square.
A. 9; (x + 3)2
B. 36; (x + 6)2
C. 9; (x – 3)2
D. 36; (x – 6)2 A B C D
Example 3

10. Solve x2 + 4x – 12 = 0 by completing the 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 = 12 Rewrite so the left side is of the form x2 + bx.
x2 + 4x + 4 = add 4 to each side.
(x + 2)2 = 16 Write the left side as a perfect square by factoring.
Example 4

11. x + 2 = ± 4 Square Root Property
Solve an Equation by Completing the Square
x + 2 = ± 4 Square Root Property
x = – 2 ± 4 Subtract 2 from each side.
x = –2 + 4 or x = –2 – 4 Write as two equations.
x = 2 x = –6 Solve each equation.
Answer: The solution set is {–6, 2}.
Example 4

12. Solve 3x2 – 2x – 1 = 0 by completing the square.
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.
Example 5

13. Equation with a ≠ 1
Write the left side as a perfect square by factoring. Simplify the right side.
Square Root Property
Example 5

14. or Write as two equations.
Equation with a ≠ 1
or Write as two equations.
x = 1 Solve each equation.
Answer:
Example 5

15. A B C D Solve 2x2 + 11x + 15 = 0 by completing the square. A. B. C. D.
Example 5

16. Solve x 2 + 4x + 11 = 0 by completing the square.
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
Example 6

17. Subtract 2 from each side.
Equation with Imaginary Solutions
Subtract 2 from each side.
Example 6

18. End of the Lesson