1. CONFIDENTIAL 1 Completing the Square Completing the Square

2. CONFIDENTIAL 2 Warm Up Solve. Round to the nearest hundredth. 1) 12 = 5x 2 2) 3x 2 - 4 = 15 3) x 2 - 7 = 19

3. CONFIDENTIAL 3 You have 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. 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.

4. CONFIDENTIAL 4 x 2 + 6x + 9 x 2 - 8x +16 6262 2 -8 2 2 Divide the coefficient of the x- term by 2, then square the result to get the constant term. 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.

5. CONFIDENTIAL 5 Completing the Square WORDSNUMBERSALGEBRA To complete the square of x 2 + bx, add to the expression. This will form a perfect square trinomial. x 2 + 6x + __ x 2 + 6x + x 2 + 6x + 9 (x + 3) 2 x 2 + bx + __ x 2 + bx + x 2 + bx + 9 (x + b) 2 2 6262 2 6262 2 b2b2 2

6. CONFIDENTIAL 6 Completing the Square Complete the square to form a perfect square trinomial. A) x 2 + 10x + __ x 2 + 10x B) x 2 - 9x + __ x 2 + 9x -9 2 2 10 2 2 = 5 2 = 25 81 4 = x 2 + 10x + 25 x 2 - 9x + 81 4 Identify b. Find b2b2 2 Add to the expression b2b2 2

7. CONFIDENTIAL 7 Now you try! Complete the square to form a perfect square trinomial. 1a. x 2 + 12x + __ 1b. x 2 - 5x + __ 1c. 8x + x 2 __

8. CONFIDENTIAL 8 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. CONFIDENTIAL 9 Solving a Quadratic Equation by Completing the Square Step1: Write the equation in the form x 2 + bx = c. Step2: Find. Step3: Complete the square by adding to both sides of the equation. Step4: Factor the perfect-square trinomial. Step5: Take the square root of both sides. Step6: Write two equations, using both the positive and negative square root, and solve each equation. b2b2 2 b2b2 2

10. CONFIDENTIAL 10 Solving + bx = c by Completing the Square Solving x 2 + bx = c by Completing the Square Solve by completing the square. A) x 2 + 14x = 15 Step1: x 2 + 14x = 15 The equation is in the form x 2 + bx = c. Step2: 14 2 2 = 7 2 = 49 Find b2b2 2 Complete the square. Step3: x 2 + 14x + 49 = 15 + 49 Factor and simplify. Step4: (x + 7) 2 = 64 Take the square root of both sides. Step5: x + 7 = ±8 Next page 

11. CONFIDENTIAL 11 Write and solve two equations. Step6: x + 7 = 8 or x + 7 = -8 x = 1 or x = -15 The solutions are 1 and -15. Check x 2 + 14x = 15 (1) 2 + 14(1) 15 1 + 14 15 15 x 2 + 14x = 15 (-15) 2 + 14(-15) 15 225 – 210 15 15

12. CONFIDENTIAL 12 B) x 2 - 2x - 2 = 0 Step1: x 2 - 2x - 2 = 0 The equation is in the form x 2 + bx = c. Step2: -2 2 2 = (-1) 2 = 1 Find b2b2 2 Complete the square. Step3: x 2 - 2x + 1 = 2 + 1 Factor and simplify. Step4: (x - 1) 2 = 3 Take the square root of both sides. Step5: x - 1 = ±√3 Write and solve two equations. Step6: x - 1 = √3 or x - 1 = - √3 x = 1 + √3 or x = 1 - √3 The solutions are 1 + √3 and 1 - √3.

13. CONFIDENTIAL 13 Now you try! Solve by completing the square. 2a. x 2 + 10x = -9 2b. t 2 - 8t - 5 = 0

14. CONFIDENTIAL 14 Solving a + bx = c by Completing the Square Solving ax 2 + bx = c by Completing the Square Solve by completing the square. A) -2x 2 + 12x - 20 = 0 Step1: -2x 2 + 12x - 20 = 0 2 2 2 Write in the form x 2 + bx = c. Step2: -6 2 2 = (-3) 2 = 9 Find b2b2 2 Divide by -2 to make a = 1. x 2 - 6x + 10 = 0 x 2 - 6x = -10 Complete the square. Step3: x 2 - 6x + 9 = -10 + 9 Factor and simplify. Step4: (x - 3) 2 = -1 There is no real number whose square is negative, so there are no real solutions.

15. CONFIDENTIAL 15 B) 3x 2 - 10x = -3 Step1: 3x 2 - 10x = -3 3 3 3 Rewrite using like denominators. Find b2b2 2 Divide by 3 to make a = 1. x 2 - 10x = -1 3 x 2 + (-10x) + 1 = 0 3 Step2: -10. 1 3 2 2 = = 100 = 25 36 9 -10 6 2 Step3: x 2 + (-10x) + 25 = -9 + 25 3 9 9 9 x 2 + (-10x) + 25 = -1 + 25 3 9 9 Complete the square. Next page 

16. CONFIDENTIAL 16 Write and solve two equations. Factor and simplify. Step5: x – 5 = ± 4 3 3 The solutions are 3 and 1. 3 Step4: x – 5 = 16 3 9 2 Step6: x – 5 = - 4 or x – 5 = - 4 3 3 3 3 Take the square root of both sides. x = 3 or x = 1 3

17. CONFIDENTIAL 17 Now you try! Solve by completing the square. 3a. 3x 2 - 5x - 2 = 0 3b. 4t 2 - 4t + 9 = 0

18. CONFIDENTIAL 18 Problem-Solving Application A landscaper is designing a rectangular brick patio. She has enough bricks to cover 144 square feet. She wants the length of the patio to be 10 feet greater than the width. What dimensions should she use for the patio? Round to the nearest hundredth of a foot. There are enough bricks to cover 144 square feet. One edge of the patio is to be 10 feet longer than the other edge. Set the formula for the area of a rectangle equal to 144, the area of the patio. Solve the equation. Let x be the width. Then x + 10 is the length.

19. CONFIDENTIAL 19 Use the formula for area of a rectangle, l × w = A (x + 10 ) x = 144 Step1: x 2 + 10x = 144 The equation is in the form x 2 + bx = c. Step2: 10 2 2 = 5 2 = 25 Find b2b2 2 Complete the square. Step3: x 2 + 10x + 24 = 144 + 25 Factor and simplify. Step4: (x + 5) 2 = 169 Take the square root of both sides. Step5: x + 5 = ±13 Next page 

20. CONFIDENTIAL 20 Write and solve two equations. Step6: x + 5 = 13 or x + 5 = -13 x = 8 or x = -18 Negative numbers are not reasonable for length, so x = 8 is the only solution that makes sense. The width is 8 feet, and the length is 8 + 10, or 18, feet. The length of the patio is 10 feet greater than the width. Also, 8 (18) = 144.

21. CONFIDENTIAL 21 Now you try! Solve using square roots. Check your answer. 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.

22. CONFIDENTIAL 22 Assessment 1) x 2 + 14x + __ Complete the square to form a perfect square trinomial. 2) x 2 - 4x + __ 3) x 2 - 3x + __

23. CONFIDENTIAL 23 Solve by completing the square. 6) x 2 + x = 30 5) x 2 - 8x = 9 4) x 2 + 6x = -5

24. CONFIDENTIAL 24 9) x 2 + 16x = 92 7) x 2 + 2x = 21 8) x 2 - 10x = -9 Solve by completing the square.

25. CONFIDENTIAL 25 10) The length of a rectangle is 4 meters longer than the width. The area of the rectangle is 80 square meters. Find the length and width. Round your answers to the nearest tenth of a meter.

26. CONFIDENTIAL 26 Completing the Square WORDSNUMBERSALGEBRA To complete the square of x 2 + bx, add to the expression. This will form a perfect square trinomial. x 2 + 6x + __ x 2 + 6x + x 2 + 6x + 9 (x + 3) 2 x 2 + bx + __ x 2 + bx + x 2 + bx + 9 (x + b) 2 2 6262 2 6262 2 b2b2 2 Let’s review

27. CONFIDENTIAL 27 Completing the Square Complete the square to form a perfect square trinomial. A) x 2 + 10x + __ x 2 + 10x B) x 2 - 9x + __ x 2 + 9x -9 2 2 10 2 2 = 5 2 = 25 81 4 = x 2 + 10x + 25 x 2 - 9x + 81 4 Identify b. Find b2b2 2 Add to the expression b2b2 2

28. CONFIDENTIAL 28 Solving a Quadratic Equation by Completing the Square Step1: Write the equation in the form x 2 + bx = c. Step2: Find. Step3: Complete the square by adding to both sides of the equation. Step4: Factor the perfect-square trinomial. Step5: Take the square root of both sides. Step6: Write two equations, using both the positive and negative square root, and solve each equation. b2b2 2 b2b2 2

29. CONFIDENTIAL 29 Solving + bx = c by Completing the Square Solving x 2 + bx = c by Completing the Square Solve by completing the square. A) x 2 + 14x = 15 Step1: x 2 + 14x = 15 The equation is in the form x 2 + bx = c. Step2: 14 2 2 = 7 2 = 49 Find b2b2 2 Complete the square. Step3: x 2 + 14x + 49 = 15 + 49 Factor and simplify. Step4: (x + 7) 2 = 64 Take the square root of both sides. Step5: x + 7 = ±8 Next page 

30. CONFIDENTIAL 30 Write and solve two equations. Step6: x + 7 = 8 or x + 7 = -8 x = 1 or x = -15 The solutions are 1 and -15. Check x 2 + 14x = 15 (1) 2 + 14(1) 15 1 + 14 15 15 x 2 + 14x = 15 (-15) 2 + 14(-15) 15 225 – 210 15 15

31. CONFIDENTIAL 31 B) x 2 - 2x - 2 = 0 Step1: x 2 - 2x - 2 = 0 The equation is in the form x 2 + bx = c. Step2: -2 2 2 = (-1) 2 = 1 Find b2b2 2 Complete the square. Step3: x 2 - 2x + 1 = 2 + 1 Factor and simplify. Step4: (x - 1) 2 = 3 Take the square root of both sides. Step5: x - 1 = ±√3 Write and solve two equations. Step6: x - 1 = √3 or x - 1 = - √3 x = 1 + √3 or x = 1 - √3 The solutions are 1 + √3 and 1 - √3.

32. CONFIDENTIAL 32 Solving a + bx = c by Completing the Square Solving ax 2 + bx = c by Completing the Square Solve by completing the square. A) -2x 2 + 12x - 20 = 0 Step1: -2x 2 + 12x - 20 = 0 2 2 2 Write in the form x 2 + bx = c. Step2: -6 2 2 = (-3) 2 = 9 Find b2b2 2 Divide by -2 to make a = 1. x 2 - 6x + 10 = 0 x 2 - 6x = -10 Complete the square. Step3: x 2 - 6x + 9 = -10 + 9 Factor and simplify. Step4: (x - 3) 2 = -1 There is no real number whose square is negative, so there are no real solutions.

33. CONFIDENTIAL 33 Problem-Solving Application A landscaper is designing a rectangular brick patio. She has enough bricks to cover 144 square feet. She wants the length of the patio to be 10 feet greater than the width. What dimensions should she use for the patio? Round to the nearest hundredth of a foot. There are enough bricks to cover 144 square feet. One edge of the patio is to be 10 feet longer than the other edge. Set the formula for the area of a rectangle equal to 144, the area of the patio. Solve the equation. Let x be the width. Then x + 10 is the length.

34. CONFIDENTIAL 34 Use the formula for area of a rectangle, l × w = A (x + 10 ) x = 144 Step1: x 2 + 10x = 144 The equation is in the form x 2 + bx = c. Step2: 10 2 2 = 5 2 = 25 Find b2b2 2 Complete the square. Step3: x 2 + 10x + 24 = 144 + 25 Factor and simplify. Step4: (x + 5) 2 = 169 Take the square root of both sides. Step5: x + 5 = ±13 Next page 

35. CONFIDENTIAL 35 Write and solve two equations. Step6: x + 5 = 13 or x + 5 = -13 x = 8 or x = -18 Negative numbers are not reasonable for length, so x = 8 is the only solution that makes sense. The width is 8 feet, and the length is 8 + 10, or 18, feet. The length of the patio is 10 feet greater than the width. Also, 8 (18) = 144.

36. CONFIDENTIAL 36 You did a great job today!