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Splash Screen. 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

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  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. 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. Equation with Rational Roots Solve x2 + 14x + 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. ? ? 12 + 14(1) + 49 = 64 (–15)2 + 14(–15) + 49 = 64 ? ? 1 + 14 + 49 = 64 225 + (–210) + 49 = 64 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. x2 + 14x + 49 = 64 x2 + 14x + 49 = 64 64 = 64 64 = 64   Example 1

  13. Solve x2 – 16x + 64 = 25 by using the Square Root Property. A. {–1, 9} B. {11, 21} C. {3, 13} D. {–13, –3} Example 1

  14. Original equation Factor the perfect square trinomial. Square Root Property Add 2 to each side. Write as two equations. Use a calculator. Equation with Irrational Roots Solve x2 – 4x + 4 = 13 by using the Square Root Property. Example 2

  15. 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 x2 – 4x + 4 = 13 Original equation x2 – 4x – 9 = 0 Subtract 13 from each side. y = x2 – 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. A. B. C. D. Solve x2 – 4x + 4 = 8 by using the Square Root Property. Example 2

  18. Concept

  19. Complete the Square Find the value of c that makes x2 + 12x + 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 1. 62 = 36 Step 3 Add the result of Step 2 to x2 + 12x + 36x2 + 12x. 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. 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. x2 + 4x+ 4 = 12 + 4 add 4 toeach side. 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. (x + 2)2 = 16 Write the left side as a perfect square by factoring. Example 4

  22. 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. A. B. C. D. Solve x2 + 6x + 8 = 0 by completing the square. Example 4

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

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

  28. Notice that x2 + 4x + 11 is not a perfect square. Rewrite so the left side is of the form x2 + bx. Since , add 4 to each side. Write the left side as a perfect square. Square Root Property Equation with Imaginary Solutions Solve x2 + 4x + 11 = 0 by completing the square. Example 6

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

  30. A. B. C. D. Solve x2 + 4x + 5 = 0 by completing the square. Example 6

  31. End of the Lesson

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