Completing the Square

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Answer Key & Explanations

Completing the Square · Grade 10 · Worksheet 2

1. Olivia is designing a parabolic water slide for a new amusement park. The height of the slide (in meters) above the ground is given by the quadratic function h(x) = -3x² + 27x - 51, where x is the horizontal distance (in meters) from the starting platform. 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 slide and the horizontal distance from the start where this maximum occurs. Then, state the coordinates of the vertex. Answer: Maximum height of 39/4 meters (or 9.75 meters) at a horizontal distance of 9/2 meters (or 4.5 meters); vertex at (9/2, 39/4) Solution: Start with the function: h(x) = -3x² + 27x - 51 Factor out -3 from the x² and x terms: h(x) = -3(x² - 9x) - 51 Complete the square inside the parentheses. Take half of -9, which is -9/2, and square it to get 81/4.
   Full step-by-step solution

   Step 1: Start with the function: h(x) = -3x² + 27x - 51 Step 2: Factor out -3 from the x² and x terms: h(x) = -3(x² - 9x) - 51 Step 3: Complete the square inside the parentheses. Take half of -9, which is -9/2, and square it to get 81/4. Step 4: Add and subtract 81/4 inside the parentheses: h(x) = -3(x² - 9x + 81/4 - 81/4) - 51 Step 5: Rewrite as: h(x) = -3[(x² - 9x + 81/4) - 81/4] - 51 Step 6: Factor the perfect square trinomial: h(x) = -3[(x - 9/2)² - 81/4] - 51 Step 7: Distribute the -3: h(x) = -3(x - 9/2)² + 243/4 - 51 Step 8: Write 51 as 204/4 to combine fractions: h(x) = -3(x - 9/2)² + 243/4 - 204/4 Step 9: Simplify: h(x) = -3(x - 9/2)² + 39/4 Step 10: The vertex form is h(x) = -3(x - 9/2)² + 39/4, so the vertex is at (9/2, 39/4). Step 11: Since the coefficient of (x - 9/2)² is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the slide is 39/4 meters (or 9.75 meters), occurring at a horizontal distance of 9/2 meters (or 4.5 meters) from the starting platform. The vertex is (9/2, 39/4).

2. A projectile's height above ground is modeled by the equation h(t) = -16t² + 96t + 112, where h is height in feet and t is time in seconds. By completing the square, determine the maximum height the projectile reaches. Answer: 256 Solution: Start with the equation: h(t) = -16t² + 96t + 112 Factor out -16 from the first two terms: h(t) = -16(t² - 6t) + 112 Complete the square inside the parentheses: t² - 6t + 9 - 9 = (t - 3)² - 9 Substitute back: h(t) = -16[(t - 3)² - 9] + 112 Distribute the -16: h(t) = -16(t - 3)² + 144 + 112…
   Full step-by-step solution

   Step 1: Start with the equation: h(t) = -16t² + 96t + 112 Step 2: Factor out -16 from the first two terms: h(t) = -16(t² - 6t) + 112 Step 3: Complete the square inside the parentheses: t² - 6t + 9 - 9 = (t - 3)² - 9 Step 4: Substitute back: h(t) = -16[(t - 3)² - 9] + 112 Step 5: Distribute the -16: h(t) = -16(t - 3)² + 144 + 112 Step 6: Combine constants: h(t) = -16(t - 3)² + 256 Step 7: The vertex form shows the maximum height occurs at the vertex (t = 3), and the maximum height is 256 feet. The answer is 256.

3. 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.

4. A parabolic satellite dish is designed to have a cross-section that follows the curve y = 3x² + 18x + 31. The receiver is placed at the focus, which is located at the vertex of the parabola. Olivia needs to find the exact coordinates of the vertex to position the receiver correctly. Rewrite the equation in vertex form y = a(x - h)² + k by completing the square, and state the coordinates of the vertex (h, k). Answer: (-3, 4) Solution: Start with y = 3x² + 18x + 31. Factor out the coefficient of x² from the x-terms: y = 3(x² + 6x) + 31. Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with y = 3x² + 18x + 31. Step 2: Factor out the coefficient of x² from the x-terms: y = 3(x² + 6x) + 31. Step 3: Complete the square inside the parentheses. Take half of the coefficient of x (which is 6), so 6/2 = 3. Square it: 3² = 9. Step 4: Add and subtract 9 inside the parentheses: y = 3(x² + 6x + 9 - 9) + 31. Step 5: Rewrite the perfect square trinomial: y = 3[(x + 3)² - 9] + 31. Step 6: Distribute the 3: y = 3(x + 3)² - 27 + 31. Step 7: Combine the constant terms: y = 3(x + 3)² + 4. Step 8: The vertex form is y = 3(x - (-3))² + 4, so the vertex (h, k) is (-3, 4). The answer is (-3, 4).

5. 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.

6. x² + 6x + 8 = 0 → (x + h)² + k = 0; h = ? Answer: 3 Solution: x^2 + 6x + 8 = 0 (x + h)^2 + k = 0 The expression (x + h)^2 expands to: x^2 + 2h x + h^2 So we need to match the coefficients of x^2 + 6x + 8 with x^2 + 2h x + h^2 + k. From x^2 + 6x + 8: Coefficient of x is 6.
   Full step-by-step solution

   We start with the equation: x^2 + 6x + 8 = 0 We want to rewrite it in the form: (x + h)^2 + k = 0 --- **Step 1: Understand the goal** The expression (x + h)^2 expands to: x^2 + 2h x + h^2 So we need to match the coefficients of x^2 + 6x + 8 with x^2 + 2h x + h^2 + k. --- **Step 2: Compare coefficients of x** From x^2 + 6x + 8: Coefficient of x is 6. From expansion: coefficient of x is 2h. So: 2h = 6 h = 3 --- **Step 3: Check constant term** From expansion: (x + h)^2 + k = x^2 + 2h x + h^2 + k. Constant term in original equation: 8. So: h^2 + k = 8. Since h = 3: 9 + k = 8 k = -1 --- **Step 4: Verify** (x + 3)^2 - 1 = x^2 + 6x + 9 - 1 = x^2 + 6x + 8. Matches the original equation. --- **Final answer:** h = 3

7. Complete the square: 2x² - 28x + 97 = 0 → (x + h)² = k, find h and k? Answer: h = -7, k = 1/2 Solution: Start with 2x² - 28x + 97 = 0 Divide both sides by 2 to simplify: x² - 14x + 97/2 = 0 Move the constant term to the right side: x² - 14x = -97/2 Take half of the coefficient of x (which is -14): -14/2 = -7 Square it: (-7)² = 49 Add 49 to both sides: x² - 14x + 49 = -97/2 + 49 Simplify the right…
   Full step-by-step solution

   Step 1: Start with 2x² - 28x + 97 = 0 Step 2: Divide both sides by 2 to simplify: x² - 14x + 97/2 = 0 Step 3: Move the constant term to the right side: x² - 14x = -97/2 Step 4: Take half of the coefficient of x (which is -14): -14/2 = -7 Step 5: Square it: (-7)² = 49 Step 6: Add 49 to both sides: x² - 14x + 49 = -97/2 + 49 Step 7: Simplify the right side: 49 = 98/2, so -97/2 + 98/2 = 1/2 Step 8: Factor the left side as a perfect square: (x - 7)² = 1/2 Step 9: This is in the form (x + h)² = k, where h = -7 and k = 1/2. The answer is h = -7, k = 1/2.

8. Complete the square: x² + 14x + 47 = 0 → (x + h)² = k, find h and k? Answer: h = 7, k = 2 Solution: Start with x² + 14x + 47 = 0 Move the constant term to the right side: x² + 14x = -47 Take half of the coefficient of x (14): 14/2 = 7 Square that result: 7² = 49 Add 49 to both sides: x² + 14x + 49 = -47 + 49 Simplify the right side: x² + 14x + 49 = 2 Factor the left side as a perfect square:…
   Full step-by-step solution

   Step 1: Start with x² + 14x + 47 = 0 Step 2: Move the constant term to the right side: x² + 14x = -47 Step 3: Take half of the coefficient of x (14): 14/2 = 7 Step 4: Square that result: 7² = 49 Step 5: Add 49 to both sides: x² + 14x + 49 = -47 + 49 Step 6: Simplify the right side: x² + 14x + 49 = 2 Step 7: Factor the left side as a perfect square: (x + 7)² = 2 Step 8: Therefore, h = 7 and k = 2.