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Warm Up Simplify. 1. 2. 19 3. 4.
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Warm Up Solve each quadratic equation by factoring. 5. x2 + 8x + 16 = 0 6. x2 – 22x = 0 7. x2 – 12x + 36 = 0 x = –4 x = 11 x = 6
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Objective Solve quadratic equations by completing the square.
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Vocabulary completing the square
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In the previous lesson, you solved quadratic equations by isolating x2 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. X2 + 6x x2 – 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.
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An expression in the form x2 + bx is not a perfect square
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.
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Example 1: Completing the Square
Complete the square to form a perfect square trinomial. A. x2 + 2x + B. x2 – 6x + x2 + 2x x2 + –6x Identify b. . x2 + 2x + 1 x2 – 6x + 9
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Check It Out! Example 1 Complete the square to form a perfect square trinomial. a. x2 + 12x + b. x2 – 5x + x2 + 12x x2 + –5x Identify b. . x2 – 6x + x2 + 12x + 36
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Check It Out! Example 1 Complete the square to form a perfect square trinomial. c. 8x + x2 + x2 + 8x Identify b. . x2 + 12x + 16
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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.
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Solving a Quadratic Equation by Completing the Square
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Example 2A: Solving x2 +bx = c
Solve by completing the square. x2 + 16x = –15 The equation is in the form x2 + bx = c. Step 1 x2 + 16x = –15 Step 2 . Step 3 x2 + 16x + 64 = – Complete the square. Step 4 (x + 8)2 = 49 Factor and simplify. Take the square root of both sides. Step 5 x + 8 = ± 7 Step 6 x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 Write and solve two equations.
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Example 2A Continued Solve by completing the square.
x2 + 16x = –15 The solutions are –1 and –15. Check x2 + 16x = –15 (–1)2 + 16(–1) –15 1 – –15 – –15 x2 + 16x = –15 (–15)2 + 16(–15) –15 225 – –15 – –15
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Example 2B: Solving x2 +bx = c
Solve by completing the square. x2 – 4x – 6 = 0 Write in the form x2 + bx = c. Step 1 x2 + (–4x) = 6 Step 2 . Step 3 x2 – 4x + 4 = 6 + 4 Complete the square. Step 4 (x – 2)2 = 10 Factor and simplify. Take the square root of both sides. Step 5 x – 2 = ± √10 Step 6 x – 2 = √10 or x – 2 = –√10 x = 2 + √10 or x = 2 – √10 Write and solve two equations.
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Example 2B Continued Solve by completing the square. The solutions are 2 + √10 and x = 2 – √10. Check Use a graphing calculator to check your answer.
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Check It Out! Example 2a Solve by completing the square. x2 + 10x = –9 The equation is in the form x2 + bx = c. Step 1 x2 + 10x = –9 Step 2 . Step 3 x2 + 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 Step 6 x + 5 = 4 or x + 5 = –4 x = –1 or x = –9 Write and solve two equations.
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Check It Out! Example 2a Continued
Solve by completing the square. x2 + 10x = –9 The solutions are –9 and –1. Check x2 + 16x = –15 (–1)2 + 16(–1) –15 1 – –15 – –15 x2 + 10x = –9 (–9)2 + 10(–9) –9 81 – –9 –9 –9
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Check It Out! Example 2b Solve by completing the square. t2 – 8t – 5 = 0 Write in the form x2 + bx = c. Step 1 t2 + (–8t) = 5 Step 2 . Step 3 t2 – 8t + 16 = Complete the square. Factor and simplify. Step 4 (t – 4)2 = 21 Take the square root of both sides. Step 5 t – 4 = ± √21 Step 6 t = 4 + √21 or t = 4 – √21 Write and solve two equations.
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Check It Out! Example 2b Continued
Solve by completing the square. The solutions are t = 4 – √21 or t = 4 + √21. Check Use a graphing calculator to check your answer.
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Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square. –3x2 + 12x – 15 = 0 Step 1 Divide by – 3 to make a = 1. x2 – 4x + 5 = 0 x2 – 4x = –5 Write in the form x2 + bx = c. x2 + (–4x) = –5 Step 2 . Step 3 x2 – 4x + 4 = –5 + 4 Complete the square.
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Example 3A Continued Solve by completing the square. –3x2 + 12x – 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.
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Example 3B: Solving ax2 + bx = c by Completing the Square
Solve by completing the square. 5x2 + 19x = 4 Step 1 Divide by 5 to make a = 1. Write in the form x2 + bx = c. . Step 2
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Example 3B Continued Solve by completing the square. Step 3 Complete the square. Rewrite using like denominators. Step 4 Factor and simplify. Take the square root of both sides. Step 5
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Example 3B Continued Solve by completing the square. Write and solve two equations. Step 6 The solutions are and –4.
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Check It Out! Example 3a Solve by completing the square. 3x2 – 5x – 2 = 0 Step 1 Divide by 3 to make a = 1. Write in the form x2 + bx = c.
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Check It Out! Example 3a Continued
Solve by completing the square. . Step 2 Step 3 Complete the square. Step 4 Factor and simplify.
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Check It Out! Example 3a Continued
Solve by completing the square. Step 5 Take the square root of both sides. Step 6 Write and solve two equations. −
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Check It Out! Example 3b Solve by completing the square. 4t2 – 4t + 9 = 0 Step 1 Divide by 4 to make a = 1. Write in the form x2 + bx = c.
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Check It Out! Example 3b Continued
Solve by completing the square. 4t2 – 4t + 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.
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Understand the Problem
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.
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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.
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Use the formula for area of a rectangle.
Example 4 Continued Solve 3 Let x be the width. Then x + 2 is the length. Use the formula for area of a rectangle. l • w = A length times width = area of room x + 2 • x = 195
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Example 4 Continued Step 1 x2 + 2x = 195 Simplify. . Step 2 Complete the square by adding 1 to both sides. Step 3 x2 + 2x + 1 = Step 4 (x + 1)2 = 196 Factor the perfect-square trinomial. Take the square root of both sides. Step 5 x + 1 = ± 14 Step 6 x + 1 = 14 or x + 1 = –14 Write and solve two equations. x = 13 or x = –15
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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 , or 15, feet. 4 Look Back The length of the room is 2 feet greater than the width. Also 13(15) = 195.
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Understand the Problem
Check It Out! Example 4 An architect designs a rectangular room with an area of 400 ft2. 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.
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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.
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Check It Out! Example 4 Continued
Solve 3 Let x be the width. Then x + 8 is the length. Use the formula for area of a rectangle. l • w = A length times width = area of room X + 8 • x = 400
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Check It Out! Example 4 Continued
Step 1 x2 + 8x = 400 Simplify. Step 2 . Step 3 x2 + 8x + 16 = Complete the square by adding 16 to both sides. Step 4 (x + 4)2 = 416 Factor the perfect-square 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 equations. x 16.4 or x –24.4
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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 approximately16.4 feet, and the length is , 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) = , which is approximately 400.
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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. 4x x = 23 81 6, –8
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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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