1. Splash Screen

2. Five-Minute Check (over Lesson 4–4) CCSS Then/Now New Vocabulary
Example 1: Equation with Rational Roots
Example 2: Equation with Irrational Roots
Key Concept: Completing the Square
Example 3: Complete the Square
Example 4: Solve an Equation by Completing the Square
Example 5: Equation with a ≠ 1
Example 6: Equation with Imaginary Solutions
Lesson Menu

3. A. 5 B. C. D.
5-Minute Check 1

4. A. B. C. D.
5-Minute Check 2

5. Simplify (5 + 7i) – (–3 + 2i). A. 2 + 9i B. 8 + 5i C. 2 – 9i
D. –8 – 5i
5-Minute Check 3

6. Solve 7x2 + 63 = 0. A. ± 5i B. ± 3i C. ± 3 D. ± 3i – 3
5-Minute Check 4

7. What are the values of x and y when (4 + 2i) – (x + yi) = (2 + 5i)?
A. x = 6, y = –7
B. x = –6, y = 7
C. x = –2, y = 3
D. x = 2, y = –3
5-Minute Check 5

8. Mathematical Practices 7 Look for and make use of structure.
Content Standards
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Mathematical Practices
7 Look for and make use of structure.
CCSS

9. You factored perfect square trinomials.
Solve quadratic equations by using the Square Root Property.
Solve quadratic equations by completing the square.
Then/Now

10. completing the square
Vocabulary

11. 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

12. 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

13. Solve x 2 – 16x + 64 = 25 by using the Square Root Property.
C. {3, 13}
D. {–13, –3}
Example 1

14. 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

15. 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

16. 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

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

18. Concept

19. 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

20. 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
Example 3

21. 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

22. 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

23. Solve x2 + 6x + 8 = 0 by completing the square.D.
Example 4

24. 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

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

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

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

28. 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

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

30. Solve x 2 + 4x + 5 = 0 by completing the square.D.
Example 6

31. End of the Lesson