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## Warm Up Simplify.

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**Warm Up**• Simplify. 19 1. 2. 3. 4.**Warm Up**• Solve each quadratic equation by factoring. • 5. x2 + 8x + 16 = 0 • 6. x2– 22x + 121 = 0 • 7. x2 – 12x + 36 = 0 x = –4 x= 11 x = 6**Objective**Solve quadratic equations by completing the square.**Vocabulary**completing the square**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 + 9 x2– 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.**An expression in the form x2 + bx is not aperfect 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.**A. x2 + 2x +**B. x2 – 6x+ . 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**b. x2 – 5x+**a. x2 + 12x + . x2 – 6x + Check It Out! Example 1 Complete the square to form a perfect square trinomial. x2 + 12x x2 + –5x Identify b. x2 + 12x + 36**c. 8x + x2 +**. Check It Out! Example 1 Complete the square to form a perfect square trinomial. x2 + 8x Identify b. x2 + 12x + 16**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.**.**Step 2 Step 6x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 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 3x2 + 16x + 64 = –15 + 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 Write and solve two equations.**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 Example 2A Continued Solve by completing the square. x2 + 16x = –15 The solutions are –1 and –15.**Step 2**. Step 5 x – 2 = ± √10 Step 6x – 2 = √10 or x – 2 = –√10 x = 2 + √10 or x = 2 – √10 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 3x2 – 4x + 4 = 6 + 4 Complete the square. Step 4 (x – 2)2 = 10 Factor and simplify. Take the square root of both sides. Write and solve two equations.**Example 2B Continued**Solve by completing the square. The solutions are2 + √10 and x = 2 – √10. CheckUse a graphing calculator to check your answer.**.**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. 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.**x2 + 16x = –15**x2 + 10x = –9 (–1)2 + 16(–1) –15 (–9)2 + 10(–9) –9 1 – 16 –15 81 – 90 –9 –15 –15 –9 –9 Check It Out! Example 2a Continued Solve by completing the square. x2 + 10x = –9 The solutions are –9 and –1. Check**.**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. 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.**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.**.**Step 1 x2 – 4x + 5 = 0 x2 – 4x = –5 Step 2 Step 3 x2 – 4x + 4 = –5 + 4 Example 3A: Solving ax2 + bx = c by Completing the Square Solve by completing the square. –3x2 + 12x – 15 = 0 Divide by – 3 to make a = 1. Write in the form x2 + bx = c. x2+ (–4x) = –5 Complete the square.**(x – 2)2= –1**Step 4 Example 3A Continued Solve by completing the square. –3x2 + 12x – 15 = 0 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.**.**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**Step 4**Step 5 Example 3B Continued Solve by completing the square. Step 3 Complete the square. Rewrite using like denominators. Factor and simplify. Take the square root of both sides.**The solutions are and –4.**Example 3B Continued Solve by completing the square. Write and solve two equations. Step 6**Check It Out! Example 3a**Solve by completing the square. 3x2 – 5x – 2 = 0 Divide by 3 to make a = 1. Step 1 Write in the form x2 + bx = c.**.**Step 3 Step 4 Check It Out! Example 3a Continued Solve by completing the square. Step 2 Complete the square. Factor and simplify.**Step 6**− Check It Out! Example 3a Continued Solve by completing the square. Step 5 Take the square root of both sides. Write and solve two equations.**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.**.**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.**1**Understand the Problem List the important information: • The room area is 195 square feet. • The width is 2 feet less than the length. 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.**Make a Plan**2 Example 4 Continued Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.**3**Solve = x + 2 x 195 • width length times area of room = Use the formula for area of a rectangle. l • w = A Example 4 Continued Let x be the width. Then x + 2 is the length.**.**Step 2 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 Step 6 x + 1 = 14 or x + 1 = –14 Write and solve two equations. x = 13 or x = –15**4**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. Look Back The length of the room is 2 feet greater than the width. Also 13(15) = 195.**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 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. The answer will be the length and width of the room.**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.**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.**.**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. 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**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 approximately16.4 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.**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**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