1. 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 Homework: Part I

2. Homework: Part II 6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft 2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?

3. Warm Up Part I Simplify. 19 1.2. 3.4.

4. Warm Up Part II Solve each quadratic equation by factoring. 5. x 2 + 8x + 16 = 0 6. x 2 – 22x + 121 = 0 7. x 2 – 12x + 36 = 0 –4 11 6

5. completing the square Vocabulary

6. In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method also works if the quadratic equation, when written in standard form, is a perfect square.

7. (x + n) 2 = x 2 + 2nx + n 2 (x – n) 2 = x 2 – 2nx + n 2 Divide the coefficient of the x-term by 2. Then square the result to get the constant term. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.

8. An expression in the form x 2 + bx is not a perfect square. However, you can use the relationship shown on the previous slide to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square.

9. Additional 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..

10. Partner Share! 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 – 5x +.

11. Partner Share! Example 1 Complete the square to form a perfect square trinomial. c. 8x + x 2 + x 2 + 8x x 2 + 8x + 16 Identify b..

12. 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.

13. Solving a Quadratic Equation by Completing the Square

14. Additional Example 2A: Solving x 2 +bx = c by Completing the Square Solve by completing the square. Check your answer. 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..

15. Additional Example 2A Continued Solve by completing the square. x 2 + 16x = –15 Check x 2 + 16x = –15 (–1) 2 + 16(–1) –15 1 – 16 –15 –15 x 2 + 16x = –15 (–15) 2 + 16(–15) –15 225 – 240 –15 –15

16. Additional Example 2B: Solving x 2 +bx = c by Completing the square Solve by completing the square. Check your answer. 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 2 Step 6 x – 2 = √10 or x – 2 = – √10 x = 2 √10 ±

17. Additional Example 2B Continued Solve by completing the square. The exact solutions are 2 + √10 and x = 2 – √10. Check Use a graphing calculator to check your answer.

18. The expressions and can be written as one expression:, which is read as “1 plus or minus the square root of 3.” Writing Math

19. Partner Share! Example 2a Solve by completing the square. Check your answer. 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..

20. Check It Out! Example 2a Continued Solve by completing the square. Check your answer. x 2 + 10x = –9 (–9) 2 + 10(–9) –9 81 – 90 –9 –9 x 2 + 10x = –9 (–1) 2 + 10(–1) –9 1 – 10 –9 –9 Check

21. Partner Share! Example 2b t 2 – 8t – 5 = 0 Step 1 t 2 + (–8t) = 5 Step 3 t 2 – 8t + 16 = 5 + 16 Complete the square. Write in the form x 2 + bx = c. Step 4 (t – 4) 2 = 21 Step 5 t – 4 = ± √21 Factor and simplify. Take the square root of both sides. Write and solve two equations. Step 6 t = 4 + √21 or t = 4 – √21. Step 2 Solve by completing the square. Check your answer.

22. Check It Out! Example 2b Continued Check Use a graphing calculator to check your answer. The exact solutions are 4 – √21 and 4 + √21. Solve by completing the square. Check your answer.

23. Additional Example 3A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. –3x 2 + 12x – 15 = 0 x 2 – 4x + 5 = 0 x 2 – 4x = –5 x 2 + (–4x) = –5 Step 3x 2 – 4x + 4 = –5 + 4 Divide both sides of the equation by – 3 so that a = 1. Write in the form x 2 + bx = c. Complete the square by adding 4 to both sides.. Step 1 Step 2

24. Additional Example 3A Continued Solve by completing the square. –3x 2 + 12x – 15 = 0 Step 4 (x – 2) 2 = –1 Factor and simplify. ø There is no real number whose square is negative, so there are no real solutions.

25. Additional Example 3B: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. 5x 2 + 19x = 4 Step 1 Divide both sides of the equation by 5 so that a = 1. Write in the form x 2 + bx = c. Step 2.

26. Step 3 Additional Example 3B Continued Solve by completing the square. Factor and simplify. Step 4 Rewrite using like denominators. Complete the square by adding to both sides.

27. Additional Example 3B Continued Solve by completing the square. Step 6 Write and solve two equations. Step 5 Take the square root of both sides.

28. Check It Out! Example 3a Solve by completing the square. Check your answer. 3x 2 – 5x – 2 = 0 Step 1 Divide both sides of the equation by 3 so that a = 1. Write in the form x 2 + bx = c.

29. Factor and simplify. Step 3 Step 4 Check It Out! Example 3a Continued Solve by completing the square. Check your answer. Step 2. Complete the square by adding to both sides.

30. Check It Out! Example 3a Continued Solve by completing the square. Check your answer. Write and solve two equations. Step 6 Take the square root of both sides. Step 5

31. Check It Out! Example 3a Continued Solve by completing the square. Check your answer. 3x 2 – 5x – 2 = 0 Check 3(2) 2 – 5(2) – 2 0 12 – 10 – 2 0 0 3x 2 – 5x – 2 = 0 0 3x 2 – 5x – 2 = 0 3 2 – 5 – 2 0

32. Partner Share! Example 3b Solve by completing the square. 4t 2 – 4t + 9 = 0 Step 1 Divide by 4 to make a = 1. Write in the form x 2 + bx = c.

33. Check It Out! Example 3b Continued Solve by completing the square. 4t 2 – 4t + 9 = 0 Step 2 Step 3 Factor and simplify. Complete the square. Step 4 There is no real number whose square is negative, so there are no real solutions..

34. Additional Example 4: Problem-Solving Application A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary. 1 Understand the Problem The answer will be the length and width of the room. List the important information: The room area is 195 square feet. The width is 2 feet less than the length.

35. 2 Make a Plan Additional Example 4 Continued Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.

36. Additional Example 4 Continued Solve 3 Let x be the length. Then x - 2 is the width. Use the formula for area of a rectangle. l w = A length times width = area of room X X - 2 = 195

37. Step 1 x 2 - 2x = 195 Step 2 Step 3 x 2 - 2x + 1 = 195 + 1 Step 4 (x - 1) 2 = 196 Simplify. Complete the square by adding 1 to both sides. Factor the perfect-square trinomial. Additional Example 4 Continued Take the square root of both sides. Step 5 x - 1 = ±14.

38. Negative numbers are not reasonable for length, so x = 15 is the only solution that makes sense. The length 15 feet, and the width is 15 - 2, or 13 feet. Additional Example 4 Continued Look Back 4 The width of the room is 2 feet less than the length. Also 15(13) = 195. Step 6 x - 1 = 14 or x - 1 = –14 Write and solve two equations. x = 15 or x = –13

39. Partner Share! Example 4 A rectangular room has an area of 400 ft 2. The length is 8 ft longer than the width. Find the dimensions of the room. Round to the nearest tenth of a foot. 1 Understand the Problem The answer will be the length and width of the room. List the important information: The room area is 400 square feet. The length is 8 feet more than the width.

40. 2 Make a Plan Set the formula for the area of a rectangle equal to 400, the area of the room. Solve the equation. Check It Out! Example 4 Continued

41. Solve 3 Let x be the width. Then x + 8 is the length. Use the formula for area of a rectangle. l w = A Check It Out! Example 4 Continued length times width = area of room X + 8 x = 400

42. Step 1 x 2 + 8x = 400 Step 3 x 2 + 8x + 16 = 400 + 16 Step 4 (x + 4) 2 = 416 Simplify. Complete the square by adding 16 to both sides. Factor the perfect-square trinomial. Step 2 Partner Share! Example 4 Continued Take the square root of both sides. Step 5 x + 4  ±20.4 Step 6 x + 4  20.4 or x + 4  –20.4 Write and solve two equations. x  16.4 or x  –24.4.

43. Negative numbers are not reasonable for length, so x  16.4 is the only solution that makes sense. The width is approximately 16.4 feet, and the length is 16.4 + 8, or approximately 24.4 feet. Check It Out! Example 4 Continued Look Back 4 The length of the room is 8 feet longer than the width. Also 16.4(24.4) = 400.16, which is approximately 400 ft 2.

44. 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 Review: Part I 81 6, –8

45. Lesson Review: Part II 6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft 2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner? 8 feet by 15 feet