9 7 Completing the Square Preview Warm Up

9 -7 Completing the Square Preview Warm Up California Standards Lesson Presentation

9 -7 Completing the Square Warm Up Part I Simplify. 1. 2. 3. 4. 19

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

9 -7 Completing the Square California Standards 14. 0 Students solve a quadratic equation by factoring or 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

9 -7 Completing the Square Vocabulary completing the square

9 -7 Completing the Square 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.

9 -7 Completing the Square When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. (x + n)2 = x 2 + 2 nx + n 2 Divide the (x – n)2 coefficient of the x = x 2 – 2 nx + n 2 -term by 2. Then square the result to get the constant term.

9 -7 Completing the Square 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 -7 Completing the Square Additional 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 x 2 – 6 x + 9

9 -7 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 x 2 – 5 x +

9 -7 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

9 -7 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.

9 -7 Completing the Square Solving a Quadratic Equation by Completing the Square

9 -7 Completing the Square Additional 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 The equation is in the form x 2 + bx = c. Step 2 . Step 3 x 2 + 16 x + 64 = – 15 + 64 Complete the square. 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 Factor and simplify. Take the square root of both sides. Write and solve two equations.

9 -7 Completing the Square Additional Example 2 A Continued Solve by completing the square. x 2 + 16 x = – 15 Check x 2 + 16 x = – 15 (– 1)2 + 16(– 1) 1 – 16 – 15 x 2 + 16 x = – 15 (– 15)2 + 16(– 15) 225 – 240 – 15

9 -7 Completing the Square Additional Example 2 B: Solving x 2 +bx = c by Completing the square 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 equations. Step 5 x – 2 = ± √ 10

9 -7 Completing the Square Additional Example 2 B 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.

9 -7 Completing the Square Writing Math The expressions and can be written as one expression: , which is read as “ 1 plus or minus the square root of 3. ”

9 -7 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 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.

9 -7 Completing the Square Check It Out! Example 2 a Continued Solve by completing the square. Check your answer. x 2 + 10 x = – 9 Check x 2 + 10 x = – 9 (– 1)2 + 10(– 1) – 9 (– 9)2 + 10(– 9) – 9 1 – 10 – 9 – 9 81 – 90 – 9 – 9

9 -7 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 Write and solve two equations.

9 -7 Completing the Square Check It Out! Example 2 b Continued Solve by completing the square. Check your answer. The exact solutions are 4 – √ 21 and 4 + √ 21. Check Use a graphing calculator to check your answer.

9 -7 Completing the Square Additional Example 3 A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. – 3 x 2 + 12 x – 15 = 0 Step 1 x 2 – 4 x + 5 = 0 x 2 – 4 x = – 5 x 2 + (– 4 x) = – 5 Step 2 Step 3 Divide both sides of the equation by – 3 so that a = 1. Write in the form x 2 + bx = c. . x 2 – 4 x + 4 = – 5 + 4 Complete the square by adding 4 to both sides.

9 -7 Completing the Square Additional 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. ø There is no real number whose square is negative, so there are no real solutions.

9 -7 Completing the Square Additional 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 both sides of the equation by 5 so that a = 1. Write in the form x 2 + bx = c. Step 2 .

9 -7 Completing the Square Additional Example 3 B Continued Solve by completing the square. Step 3 Complete the square by adding to both sides. Rewrite using like denominators. Step 4 Factor and simplify.

9 -7 Completing the Square Additional Example 3 B Continued Solve by completing the square. Step 5 Step 6 Take the square root of both sides. Write and solve two equations.

9 -7 Completing the Square Check It Out! Example 3 a Solve by completing the square. Check your answer. 3 x 2 – 5 x – 2 = 0 Step 1 Divide both sides of the equation by 3 so that a = 1. Write in the form x 2 + bx = c.

9 -7 Completing the Square Check It Out! Example 3 a Continued Solve by completing the square. Check your answer. Step 2 . Step 3 Complete the square by adding to both sides. Step 4 Factor and simplify.

9 -7 Completing the Square Check It Out! Example 3 a Continued Solve by completing the square. Check your answer. Step 5 Take the square root of both sides. Step 6 Write and solve two equations.

9 -7 Completing the Square Check It Out! Example 3 a Continued Solve by completing the square. Check your answer. 3 x 2 – 5 x – 2 = 0 Check 3 x 2 – 5 x – 2 = 0 3(2)2 – 5(2) – 2 12 – 10 – 2 0 0 3 x 2 – 5 x – 2 = 0 3 2 – 5 – 2 0 0 0

9 -7 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.

9 -7 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.

9 -7 Completing the Square 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.

9 -7 Completing the Square Additional 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.

9 -7 Completing the Square Additional 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 • length times x+2 • w width x = A = area of room = 195

9 -7 Completing the Square Additional 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. Step 5 x + 1 = ± 14 Take the square root of both sides.

9 -7 Completing the Square 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. 4 Look Back The length of the room is 2 feet greater than the width. Also 13(15) = 195.

9 -7 Completing the Square Check It Out! 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.

9 -7 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.

9 -7 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 • times • w = width = area of room x = A 400

9 -7 Completing the Square Check It Out! Example 4 Continued Step 1 x 2 + 8 x = 400 Simplify. . Step 2 Step 3 x 2 + 8 x + 16 = 400 + 16 Complete the square by Step 4 (x + 4)2 = 416 Step 5 x + 4 ± 20. 4 adding 16 to both sides. Factor the perfect-square trinomial. Take the square root of both sides. Step 6 x + 4 20. 4 or x + 4 – 20. 4 x 16. 4 or x – 24. 4 Write and solve two equations.

9 -7 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 ft 2.

9 -7 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 6, – 8

9 -7 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