1. Factoring Polynomials

36. Holt Algebra 1 9-8 Completing the Square 9-8 Completing the Square Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

37. Holt Algebra 1 9-8 Completing the Square Warm Up Simplify. 19 1.2. 3.4.

38. Holt Algebra 1 9-8 Completing the Square Warm Up Solve each quadratic equation by factoring. 5. x 2 + 8x + 16 = 0 6. x 2 – 22x + 121 = 0 7. x 2 – 12x + 36 = 0 x = –4 x = 11 x = 6

39. Holt Algebra 1 9-8 Completing the Square Solve quadratic equations by completing the square. Objective

40. Holt Algebra 1 9-8 Completing the Square An expression in the form x 2 + bx is not a perfect square. However, you can use the algorithm below to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square.

41. Holt Algebra 1 9-8 Completing the Square Example 1: Completing the Square Complete the square to form a perfect square trinomial. A. x 2 + 2x + B. x 2 – 6x + x 2 + 2x x 2 + 2x + 1 x 2 + –6x x 2 – 6x + 9 Identify b..

42. Holt Algebra 1 9-8 Completing the Square Check It Out! Example 1 Complete the square to form a perfect square trinomial. a. x 2 + 12x + b. x 2 – 5x + x 2 + 12x x 2 + 12x + 36 x 2 + –5x Identify b. x 2 – 6x +.

43. Holt Algebra 1 9-8 Completing the Square Check It Out! Example 1 Complete the square to form a perfect square trinomial. c. 8x + x 2 + x 2 + 8x x 2 + 12x + 16 Identify b..

44. Holt Algebra 1 9-8 Completing the Square To solve a quadratic equation in the form x 2 + bx = c, first complete the square of x 2 + bx. Then you can solve using square roots.

45. Holt Algebra 1 9-8 Completing the Square Solving a Quadratic Equation by Completing the Square

46. Holt Algebra 1 9-8 Completing the Square Example 2A: Solving x 2 +bx = c Solve by completing the square. x 2 + 16x = –15 Step 1 x 2 + 16x = –15 Step 2 Step 3 x 2 + 16x + 64 = –15 + 64 Step 4 (x + 8) 2 = 49 Step 5 x + 8 = ± 7 Step 6 x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 The equation is in the form x 2 + bx = c. Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations..

47. Holt Algebra 1 9-8 Completing the Square Example 2B: Solving x 2 +bx = c Solve by completing the square. x 2 – 4x – 6 = 0 Step 1 x 2 + (–4x) = 6 Step 3 x 2 – 4x + 4 = 6 + 4 Step 4 (x – 2) 2 = 10 Step 5 x – 2 = ± √10 Write in the form x 2 + bx = c. Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations. Step 6 x – 2 = √10 or x – 2 = – √10 x = 2 + √10 or x = 2 – √ 10. Step 2

48. Holt Algebra 1 9-8 Completing the Square Check It Out! Example 2a Solve by completing the square. x 2 + 10x = –9 Step 1 x 2 + 10x = –9 Step 3 x 2 + 10x + 25 = –9 + 25 Complete the square. The equation is in the form x 2 + bx = c. Step 2 Step 4 (x + 5) 2 = 16 Step 5 x + 5 = ± 4 Step 6 x + 5 = 4 or x + 5 = –4 x = –1 or x = –9 Factor and simplify. Take the square root of both sides. Write and solve two equations..

49. Holt Algebra 1 9-8 Completing the Square Example 3A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. –3x 2 + 12x – 15 = 0 Step 1 x 2 – 4x + 5 = 0 x 2 – 4x = –5 x 2 + (–4x) = –5 Step 3x 2 – 4x + 4 = –5 + 4 Divide by – 3 to make a = 1. Write in the form x 2 + bx = c. Complete the square.. Step 2

50. Holt Algebra 1 9-8 Completing the Square Example 3A Continued Solve by completing the square. –3x 2 + 12x – 15 = 0 Step 4 (x – 2) 2 = –1 There is no real number whose square is negative, so there are no real solutions. Factor and simplify. Step 3x 2 – 4x + 4 = –5 + 4

51. Holt Algebra 1 9-8 Completing the Square Complete the square to form a perfect square trinomial. 1. x 2 +11x + 2. x 2 – 18x + Solve by completing the square. 3. x 2 – 2x – 1 = 0 4. 3x 2 + 6x = 144 5. 4x 2 + 44x = 23 Lesson Quiz: Part I 81 6, –8