Completing the Square

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Completing the Square · Grade 10 · Worksheet 2

1. Charlotte is designing a parabolic arch for a new entrance to a botanical garden. The height of the arch (in meters) above the ground is given by the quadratic function h(x) = -2x^2 + 28x - 78, where x is the horizontal distance (in meters) from the left base of the arch. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to determine the maximum height of the arch and the horizontal distance from the left base where this maximum occurs. Then, state the coordinates of the vertex. Answer: Maximum height of 20 meters at a horizontal distance of 7 meters; vertex at (7, 20) Solution: Start with the function: h(x) = -2x^2 + 28x - 78 Factor out -2 from the x^2 and x terms: h(x) = -2(x^2 - 14x) - 78 Complete the square inside the parentheses. Take half of -14, which is -7, and square it to get 49.
   Full step-by-step solution

   Step 1: Start with the function: h(x) = -2x^2 + 28x - 78 Step 2: Factor out -2 from the x^2 and x terms: h(x) = -2(x^2 - 14x) - 78 Step 3: Complete the square inside the parentheses. Take half of -14, which is -7, and square it to get 49. Step 4: Add and subtract 49 inside the parentheses: h(x) = -2(x^2 - 14x + 49 - 49) - 78 Step 5: Rewrite as: h(x) = -2[(x^2 - 14x + 49) - 49] - 78 Step 6: Factor the perfect square trinomial: h(x) = -2[(x - 7)^2 - 49] - 78 Step 7: Distribute the -2: h(x) = -2(x - 7)^2 + 98 - 78 Step 8: Simplify: h(x) = -2(x - 7)^2 + 20 Step 9: The vertex form is h(x) = -2(x - 7)^2 + 20, so the vertex is at (7, 20). Step 10: Since the coefficient of (x - 7)^2 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the arch is 20 meters, occurring at a horizontal distance of 7 meters from the left base. The vertex is (7, 20).

2. Rewrite 2x² - 12x + 5 in the form a(x - h)² + k, and state the values of a, h, and k. Answer: a = 2, h = 3, k = -13 Solution: Start with 2x² - 12x + 5. Factor out 2 from the x² and x terms: 2(x² - 6x) + 5. Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with 2x² - 12x + 5. Step 2: Factor out 2 from the x² and x terms: 2(x² - 6x) + 5. Step 3: Complete the square inside the parentheses. Take half of -6: -6/2 = -3. Square it: (-3)² = 9. Step 4: Add and subtract 9 inside the parentheses: 2(x² - 6x + 9 - 9) + 5. Step 5: Rewrite as: 2[(x - 3)² - 9] + 5. Step 6: Distribute the 2: 2(x - 3)² - 18 + 5. Step 7: Combine constants: 2(x - 3)² - 13. Step 8: The vertex form is 2(x - 3)² - 13, so a = 2, h = 3, k = -13.

3. Complete the square: 2x² - 16x + 10 = 0 → (x + a)² = b, find a and b? Answer: a = -4, b = 11 Solution: Start with 2x² - 16x + 10 = 0 Factor out 2 from the x² and x terms: 2(x² - 8x) + 10 = 0 Move the constant term to the other side: 2(x² - 8x) = -10 Take half of the coefficient of x inside the parentheses: -8/2 = -4 Square it: (-4)² = 16 Add 16 inside the parentheses, but since it is multiplied…
   Full step-by-step solution

   Step 1: Start with 2x² - 16x + 10 = 0 Step 2: Factor out 2 from the x² and x terms: 2(x² - 8x) + 10 = 0 Step 3: Move the constant term to the other side: 2(x² - 8x) = -10 Step 4: Take half of the coefficient of x inside the parentheses: -8/2 = -4 Step 5: Square it: (-4)² = 16 Step 6: Add 16 inside the parentheses, but since it is multiplied by 2, add 2*16 = 32 to the right side: 2(x² - 8x + 16) = -10 + 32 Step 7: Simplify: 2(x - 4)² = 22 Step 8: Divide both sides by 2: (x - 4)² = 11 Step 9: This is in the form (x + a)² = b, so a = -4 and b = 11.

4. Complete the square: x² + 10x + 18 = 0 → (x + h)² = k, find h and k? Answer: h=5, k=7 Solution: Start with x² + 10x + 18 = 0. Move the constant term to the right side: x² + 10x = -18. Take half of the coefficient of x (10): 10/2 = 5.
   Full step-by-step solution

   Step 1: Start with x² + 10x + 18 = 0. Step 2: Move the constant term to the right side: x² + 10x = -18. Step 3: Take half of the coefficient of x (10): 10/2 = 5. This is h. Step 4: Square this value: 5² = 25. Step 5: Add 25 to both sides: x² + 10x + 25 = -18 + 25. Step 6: Simplify the right side: -18 + 25 = 7. Step 7: The left side is a perfect square: (x + 5)² = 7. Step 8: Therefore, h = 5 and k = 7.

5. A rectangular garden has an area of 96 square meters. The length is 4 meters more than the width. Liam wants to build a stone path of uniform width around the entire garden that will double the total area including the path. Using completing the square method, determine the width of the stone path in meters. Answer: 2 Solution: Let’s go step-by-step. Let width of garden = \( w \) meters Length of garden = \( w + 4 \) meters Area of garden = \( w(w + 4) = 96 \) \( w^2 + 4w = 96 \) \( w^2 + 4w - 96 = 0 \) \( w^2 + 4w + 4 = 96 + 4 \) \( (w + 2)^2 = 100 \) \( w + 2 = 10 \) (positive since width > 0) \( w = 8 \) Width = 8…
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Define variables for the garden** Let width of garden = \( w \) meters Length of garden = \( w + 4 \) meters Area of garden = \( w(w + 4) = 96 \) So: \( w^2 + 4w = 96 \) --- **Step 2: Solve for \( w \)** \( w^2 + 4w - 96 = 0 \) Using completing the square: \( w^2 + 4w + 4 = 96 + 4 \) \( (w + 2)^2 = 100 \) \( w + 2 = 10 \) (positive since width > 0) \( w = 8 \) Thus: Width = 8 m, Length = 12 m. --- **Step 3: Define path width and new dimensions** Let path width = \( x \) meters (uniform around garden). New length including path = \( 12 + 2x \) New width including path = \( 8 + 2x \) --- **Step 4: Set up area equation with path** Total area (garden + path) = \( (12 + 2x)(8 + 2x) \) We are told this equals double the garden’s area. Garden area = 96, so double = 192. So: \( (12 + 2x)(8 + 2x) = 192 \) --- **Step 5: Expand and simplify** \( 96 + 24x + 16x + 4x^2 = 192 \) \( 96 + 40x + 4x^2 = 192 \) \( 4x^2 + 40x + 96 - 192 = 0 \) \( 4x^2 + 40x - 96 = 0 \) Divide by 4: \( x^2 + 10x - 24 = 0 \) --- **Step 6: Solve \( x^2 + 10x - 24 = 0 \) by completing the square** \( x^2 + 10x = 24 \) Add \( (10/2)^2 = 25 \) to both sides: \( x^2 + 10x + 25 = 24 + 25 \) \( (x + 5)^2 = 49 \) \( x + 5 = 7 \) (positive width) \( x = 2 \) --- **Step 7: Conclusion** The width of the stone path is 2 meters. --- **Final answer:** 2

6. x² - 8x + 13 = 0 → (x - h)² = k, find k? Answer: 3 Solution: Start with x² - 8x + 13 = 0 Move the constant term to the other side: x² - 8x = -13 Take half of the coefficient of x: (-8)/2 = -4 Square this result: (-4)² = 16 Add 16 to both sides: x² - 8x + 16 = -13 + 16 The left side becomes a perfect square: (x - 4)² = 3 Compare with (x - h)² = k, so k = 3…
   Full step-by-step solution

   Step 1: Start with x² - 8x + 13 = 0 Step 2: Move the constant term to the other side: x² - 8x = -13 Step 3: Take half of the coefficient of x: (-8)/2 = -4 Step 4: Square this result: (-4)² = 16 Step 5: Add 16 to both sides: x² - 8x + 16 = -13 + 16 Step 6: The left side becomes a perfect square: (x - 4)² = 3 Step 7: Compare with (x - h)² = k, so k = 3 The answer is 3.

7. A rectangular park has a length that is 10 meters longer than its width. The area of the park is 375 square meters. By completing the square, find the width of the park in meters. Answer: 15 Solution: Let the width be w meters. Then the length is w + 10 meters. Area = length × width = (w + 10)w = 375.
   Full step-by-step solution

   Step 1: Let the width be w meters. Then the length is w + 10 meters. Step 2: Area = length × width = (w + 10)w = 375. Step 3: Expand: w² + 10w = 375. Step 4: Rearrange to standard form: w² + 10w - 375 = 0. Step 5: Move the constant to the other side: w² + 10w = 375. Step 6: Complete the square: Take half of 10, which is 5, and square it to get 25. Add 25 to both sides: w² + 10w + 25 = 375 + 25. Step 7: Simplify: w² + 10w + 25 = 400. Step 8: Factor the left side as a perfect square: (w + 5)² = 400. Step 9: Take the square root of both sides: w + 5 = ±20. Step 10: Solve for w: w = -5 + 20 = 15 or w = -5 - 20 = -25. Step 11: Since width cannot be negative, w = 15. The width of the park is 15 meters.

8. Complete the square: x² - 12x + 35 = 0 → (x - h)² = k, find h and k? Answer: h=6, k=1 Solution: Start with x² - 12x + 35 = 0 Move the constant term to the right side: x² - 12x = -35 Take half of the coefficient of x: -12/2 = -6.
   Full step-by-step solution

   Step 1: Start with x² - 12x + 35 = 0 Step 2: Move the constant term to the right side: x² - 12x = -35 Step 3: Take half of the coefficient of x: -12/2 = -6. Square it: (-6)² = 36 Step 4: Add 36 to both sides: x² - 12x + 36 = -35 + 36 Step 5: Simplify the right side: -35 + 36 = 1 Step 6: The left side is a perfect square: (x - 6)² = 1 Step 7: Therefore, h = 6 and k = 1.