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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Part I Simplify. 19. 1. 2. 3. 4. Warm Up Part II Solve each quadratic equation by factoring. 5 . x 2 + 8 x + 16 = 0 6. x 2 – 22 x + 121 = 0 7. x 2 – 12 x + 36 = 0. –4. 11. 6. California

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Part I Simplify. 19 1. 2. 3. 4.

  3. Warm Up Part II • Solve each quadratic equation by factoring. • 5. x2 + 8x + 16 = 0 • 6. x2– 22x + 121 = 0 • 7. x2 – 12x + 36 = 0 –4 11 6

  4. California Standards 14.0 Students solve a quadratic equation by factoringor completing the square. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Also covered: 2.0

  5. Vocabulary completing the square

  6. In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method also works if the quadratic equation, when written in standard form, is a perfect square.

  7. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. Divide the coefficient of the x-term by 2. Then square the result to get the constant term. (x – n)2 (x + n)2 = x2 + 2nx + n2 = x2 – 2nx + n2

  8. An expression in the form x2 + bx is not aperfect square. However, you can use the relationship shown on the previous slide to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

  9. A. x2 + 2x + B. x2 – 6x+ . Additional Example 1: Completing the Square Complete the square to form a perfect square trinomial. x2 + 2x x2 + –6x Identify b. x2 + 2x + 1 x2 – 6x + 9

  10. b. x2 – 5x+ a. x2 + 12x + . x2 – 5x + Check It Out! Example 1 Complete the square to form a perfect square trinomial. x2 + 12x x2 + –5x Identify b. x2 + 12x + 36

  11. c. 8x + x2 + . Check It Out! Example 1 Complete the square to form a perfect square trinomial. x2 + 8x Identify b. x2 + 8x + 16

  12. To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

  13. Solving a Quadratic Equation by Completing the Square

  14. . Step 2 Step 6x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 Additional Example 2A: Solving x2 +bx = c by Completing the Square Solve by completing the square. Check your answer. x2 + 16x = –15 The equation is in the form x2 + bx = c. Step 1 x2 + 16x = –15 Step 3x2 + 16x + 64 = –15 + 64 Complete the square. Step 4 (x + 8)2 = 49 Factor and simplify. Step 5 x + 8 = ± 7 Take the square root of both sides. Write and solve two equations.

  15. x2 + 16x = –15 Check x2 + 16x = –15 (–15)2 + 16(–15) –15 (–1)2 + 16(–1) –15 225 – 240 –15 1 – 16 –15   –15 –15 –15 –15 Additional Example 2A Continued Solve by completing the square. x2 + 16x = –15

  16. Step 2 . Step 5 x – 2 = ± √10 Step 6x – 2 = √10 or x – 2 = –√10 ± x = 2 √10 Additional Example 2B: Solving x2 +bx = c by Completing the square Solve by completing the square. Check your answer. x2 – 4x – 6 = 0 Write in the form x2 + bx = c. Step 1 x2 + (–4x) = 6 Complete the square. Step 3x2 – 4x + 4 = 6 + 4 Factor and simplify. Step 4 (x – 2)2 = 10 Take the square root of both sides. Write and solve two equations.

  17. Additional Example 2B Continued Solve by completing the square. The exact solutions are2 + √10 and x = 2 – √10. CheckUse a graphing calculator to check your answer.

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

  19. . Step 2 Step 6x + 5 = 4 or x + 5 = –4 x = –1 or x = –9 Check It Out! Example 2a Solve by completing the square. Check your answer. x2 + 10x = –9 The equation is in the form x2 + bx = c. Step 1 x2 + 10x = –9 Step 3x2 + 10x + 25 = –9 + 25 Complete the square. Factor and simplify. Step 4 (x + 5)2 = 16 Take the square root of both sides. Step 5 x + 5 = ± 4 Write and solve two equations.

  20. x2 + 10x = –9 x2 + 10x = –9 (–1)2 + 10(–1) –9 (–9)2 + 10(–9) –9 1 – 10 –9 81 – 90 –9   –9 –9 –9 –9 Check It Out! Example 2a Continued Solve by completing the square. Check your answer. x2 + 10x = –9 Check

  21. . Step 2 Step 5 t – 4 = ± √21 Step 6t = 4 + √21 or t = 4 – √21 Check It Out! Example 2b Solve by completing the square. Check your answer. t2 – 8t – 5 = 0 Write in the form x2 + bx = c. Step 1 t2 + (–8t) = 5 Step 3t2 – 8t + 16 = 5 + 16 Complete the square. Factor and simplify. Step 4 (t – 4)2 = 21 Take the square root of both sides. Write and solve two equations.

  22. The exact solutions are4 – √21 and 4 + √21. Check It Out! Example 2b Continued Solve by completing the square. Check your answer. Check Use a graphing calculator to check your answer.

  23. Step 1 . x2 – 4x + 5 = 0 x2 – 4x = –5 Step 2 Step 3 x2 – 4x + 4 = –5 + 4 Additional Example 3A: Solving ax2 + bx = c by Completing the Square Solve by completing the square. –3x2 + 12x – 15 = 0 Divide both sides of the equation by – 3 so that a = 1. Write in the form x2 + bx = c. x2+ (–4x) = –5 Complete the square by adding 4 to both sides.

  24. (x – 2)2= –1 Step 4 Additional Example 3A Continued Solve by completing the square. –3x2 + 12x – 15 = 0 Factor and simplify. ø There is no real number whose square is negative, so there are no real solutions.

  25. . Additional Example 3B: Solving ax2 + bx = c by Completing the Square Solve by completing the square. 5x2 + 19x = 4 Step 1 Divide both sides of the equation by 5 so that a = 1. Write in the form x2 + bx = c. Step 2

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

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

  28. Check It Out! Example 3a Solve by completing the square. Check your answer. 3x2 – 5x – 2 = 0 Divide both sides of the equation by 3 so that a = 1. Step 1 Write in the form x2 + bx = c.

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

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

  31. 3x2 – 5x – 2 = 0 3x2 – 5x – 2 = 0 3(2)2 – 5(2) – 2 0 3 2 – 5– 2 0 12 – 10 – 2 0  0 0  0 0 Check It Out! Example 3a Continued Solve by completing the square. Check your answer. 3x2 – 5x – 2 = 0 Check

  32. Check It Out! Example 3b Solve by completing the square. 4t2 – 4t + 9 = 0 Divide by 4 to make a = 1. Step 1 Write in the form x2 + bx = c.

  33. . Step 3 Check It Out! Example 3b Continued Solve by completing the square. 4t2 – 4t + 9 = 0 Step 2 Complete the square. Step 4 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.

  34. 1 Understand the Problem List the important information: • The room area is 195 square feet. • The width is 2 feet less than the length. 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. The answer will be the length and width of the room.

  35. Make a Plan 2 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. 3 Solve Let x be the width. Then x + 2 is the length. = x + 2 x 195 • Use the formula for area of a rectangle. width length times area of room = Additional Example 4 Continued l • w = A

  37. . Step 2 Additional Example 4 Continued Step 1 x2 + 2x = 195 Simplify. Complete the square by adding 1 to both sides. Step 3 x2 + 2x+ 1 = 195 + 1 Factor the perfect-square trinomial. Step 4 (x + 1)2 = 196 Take the square root of both sides. Step 5 x + 1 = ±14

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

  39. 1 Understand the Problem List the important information: • The room area is 400 square feet. • The length is 8 feet more than the width. Check It Out! Example 4 A rectangular room has an area of 400 ft2. The length is 8 ft longer than the width. Find the dimensions of the room. Round to the nearest tenth of a foot. The answer will be the length and width of the room.

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

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

  42. . Step 2 Check It Out! Example 4 Continued Step 1 x2 + 8x = 400 Simplify. Step 3 x2 + 8x+ 16 = 400 + 16 Complete the square by adding 16 to both sides. Factor the perfect-square trinomial. Step 4 (x + 4)2 = 416 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. 4 Check It Out! Example 4 Continued 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. Look Back The length of the room is 8 feet longer than the width. Also 16.4(24.4) = 400.16, which is approximately 400 ft2.

  44. Lesson Quiz: Part I Complete the square to form a perfect square trinomial. 1. x2 + 11x + 2. x2 – 18x + Solve by completing the square. 3. x2 – 2x –1 = 0 4. 3x2 + 6x= 144 5. 4x2 + 44x = 23 81 6, –8

  45. Lesson Quiz: Part II 6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. 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

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