8 8 Completingthe the Square Warm Up Lesson

8 -8 Completingthe the. Square Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 1 Algebra 11 Holt Mc. Dougal

8 -8 Completing the Square Warm Up Simplify. 1. 2. 3. 4. Holt Mc. Dougal Algebra 1 19

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

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

8 -8 Completing the Square Vocabulary completing the square Holt Mc. Dougal Algebra 1

8 -8 Completing the Square In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. X 2 + 6 x + 9 Holt Mc. Dougal Algebra 1 x 2 – 8 x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.

8 -8 Completing the Square An expression in the form x 2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square. Holt Mc. Dougal Algebra 1

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

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

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

8 -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. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Solving a Quadratic Equation by Completing the Square Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 2 A: Solving x 2 +bx = c by Completing the Square Solve by completing the square. Check your answer. x 2 + 16 x = – 15 Step 1 x 2 + 16 x = – 15 Step 2 The equation is in the form x 2 + bx = c. . Step 3 x 2 + 16 x + 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. Step 6 x + 8 = 7 or x + 8 = – 7 x = – 1 or x = – 15 Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 2 A Continued Solve by completing the square. Check your answer. x 2 + 16 x = – 15 The solutions are – 1 and – 15. Check x 2 + 16 x = – 15 (– 1)2 + 16(– 1) 1 – 16 – 15 Holt Mc. Dougal Algebra 1 – 15 x 2 + 16 x = – 15 (– 15)2 + 16(– 15) 225 – 240 – 15

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

8 -8 Completing the Square Example 2 B Continued Solve by completing the square. Check your answer. The solutions are 2 + √ 10 and x = 2 – √ 10. Check Use a graphing calculator to check your answer. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 2 a Solve by completing the square. Check your answer. x 2 + 10 x = – 9 Step 1 x 2 + 10 x = – 9 Step 2 Step 3 x 2 + 10 x + 25 = – 9 + 25 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 Holt Mc. Dougal Algebra 1 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.

8 -8 Completing the Square Check It Out! Example 2 a Continued Solve by completing the square. Check your answer. x 2 + 10 x = – 9 The solutions are – 9 and – 1. Check x 2 + 10 x = – 9 (– 1)2 + 10(– 1) – 9 (– 9)2 + 10(– 9) – 9 1 – 10 – 9 – 9 81 – 90 – 9 – 9 Holt Mc. Dougal Algebra 1

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

8 -8 Completing the Square Check It Out! Example 2 b Continued Solve by completing the square. Check your answer. The solutions are t = 4 – √ 21 or t = 4 + √ 21. Check Use a graphing calculator to check your answer. Holt Mc. Dougal Algebra 1

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

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

8 -8 Completing the Square Example 3 B: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. 5 x 2 + 19 x = 4 Step 1 Divide by 5 to make a = 1. Write in the form x 2 + bx = c. Step 2 Holt Mc. Dougal Algebra 1 .

8 -8 Completing the Square Example 3 B Continued Solve by completing the square. Step 3 Complete the square. Rewrite using like denominators. Step 4 Factor and simplify. Step 5 Take the square root of both sides. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 3 B Continued Solve by completing the square. Write and solve two equations. Step 6 The solutions are Holt Mc. Dougal Algebra 1 and – 4.

8 -8 Completing the Square Check It Out! Example 3 a Solve by completing the square. 3 x 2 – 5 x – 2 = 0 Step 1 Divide by 3 to make a = 1. Write in the form x 2 + bx = c. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 3 a Continued Solve by completing the square. Step 2 . Step 3 Complete the square. Step 4 Factor and simplify. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 3 a Continued Solve by completing the square. Step 5 Take the square root of both sides. Step 6 Write and solve two equations. − Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 3 b Solve by completing the square. 4 t 2 – 4 t + 9 = 0 Step 1 Divide by 4 to make a = 1. Write in the form x 2 + bx = c. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 3 b Continued Solve by completing the square. 4 t 2 – 4 t + 9 = 0 Step 2 . Step 3 Complete the square. Step 4 Factor and simplify. There is no real number whose square is negative, so there are no real solutions. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square 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. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 4 Continued 2 Make a Plan Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 4 Continued 3 Solve Let x be the width. Then x + 2 is the length. Use the formula for area of a rectangle. l • w = A length x+2 Holt Mc. Dougal Algebra 1 times • width x = area of room = 195

8 -8 Completing the Square Example 4 Continued Step 1 x 2 + 2 x = 195 Step 2 Simplify. . Step 3 x 2 + 2 x + 1 = 195 + 1 Complete the square by adding 1 to both sides. Step 4 (x + 1)2 = 196 Factor the perfect-square trinomial. Take the square root of Step 5 x + 1 = ± 14 both sides. Step 6 x + 1 = 14 or x + 1 = – 14 Write and solve two equations. x = 13 or x = – 15 Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Example 4 Continued 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. 4 Look Back The length of the room is 2 feet greater than the width. Also 13(15) = 195. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 4 An architect designs a rectangular room with an area of 400 ft 2. The length is to be 8 ft longer than the width. Find the dimensions of the room. Round your answers 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. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 4 Continued 2 Make a Plan Set the formula for the area of a rectangle equal to 400, the area of the room. Solve the equation. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square Check It Out! Example 4 Continued 3 Solve Let x be the width. Then x + 8 is the length. Use the formula for area of a rectangle. l length X+8 Holt Mc. Dougal Algebra 1 • times • w = width = area of room x = A 400

8 -8 Completing the Square Check It Out! Example 4 Continued Step 1 x 2 + 8 x = 400 Step 2 Simplify. . Step 3 x 2 + 8 x + 16 = 400 + 16 Complete the square by adding 16 to both sides. Step 4 (x + 4)2 = 416 Factor the perfectsquare trinomial. Step 5 x + 4 ± 20. 4 Take the square root of both sides. Step 6 x + 4 20. 4 or x + 4 – 20. 4 Write and solve two x 16. 4 or x – 24. 4 equations. Holt Mc. Dougal Algebra 1

8 -8 Completing the Square 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. 4 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. Holt Mc. Dougal Algebra 1

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

8 -8 Completing the Square Lesson Quiz: 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 Holt Mc. Dougal Algebra 1