Completing the Square

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Answer Key & Explanations

Completing the Square · Grade 10 · Worksheet 1

1. Complete the square: x² + 16x + 61 = 0 → (x + a)² = b, find a and b? Answer: a = 8, b = 3 Solution: Start with x² + 16x + 61 = 0. Move the constant term to the right side: x² + 16x = -61. Take half of the coefficient of x (which is 16): 16/2 = 8.
   Full step-by-step solution

   Step 1: Start with x² + 16x + 61 = 0. Step 2: Move the constant term to the right side: x² + 16x = -61. Step 3: Take half of the coefficient of x (which is 16): 16/2 = 8. Step 4: Square this value: 8² = 64. Step 5: Add 64 to both sides: x² + 16x + 64 = -61 + 64. Step 6: The left side is now a perfect square: (x + 8)² = 3. Step 7: Therefore, a = 8 and b = 3. The answer is a = 8, b = 3.

2. 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.

3. Noah is designing a parabolic arch for a bridge. The arch is drawn on a coordinate plane, with the base of the arch on the x-axis from x = 0 to x = 6. The height of the arch at any point x is given by the quadratic function h(x) = -x² + 6x. By completing the square, find the vertex form of this function and determine the maximum height of the arch. Answer: 9 Solution: Start with the function h(x) = -x² + 6x. Factor out -1 from the x terms: h(x) = -1(x² - 6x). Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with the function h(x) = -x² + 6x. Step 2: Factor out -1 from the x terms: h(x) = -1(x² - 6x). Step 3: Complete the square inside the parentheses. Take half of the coefficient of x, which is -6, so half is -3. Square it to get 9. Step 4: Add and subtract 9 inside the parentheses: h(x) = -1(x² - 6x + 9 - 9). Step 5: Rewrite the perfect square trinomial: x² - 6x + 9 = (x - 3)². So h(x) = -1[(x - 3)² - 9]. Step 6: Distribute the -1: h(x) = -(x - 3)² + 9. Step 7: The vertex form is h(x) = -(x - 3)² + 9. The maximum height occurs when (x - 3)² = 0, so the maximum height is 9 units. The answer is 9.

4. Complete the square: x² - 9x + 15 = 0 → (x + h)² = k, find h and k? Answer: h = -9/2, k = 21/4 Solution: Start with x² - 9x + 15 = 0 Move the constant term to the right side: x² - 9x = -15 Take half of the coefficient of x: -9/2 Square it: (-9/2)² = 81/4 Add 81/4 to both sides: x² - 9x + 81/4 = -15 + 81/4 Write the left side as a perfect square: (x - 9/2)² = -15 + 81/4 Combine the right side: -15 =…
   Full step-by-step solution

   Step 1: Start with x² - 9x + 15 = 0 Step 2: Move the constant term to the right side: x² - 9x = -15 Step 3: Take half of the coefficient of x: -9/2 Step 4: Square it: (-9/2)² = 81/4 Step 5: Add 81/4 to both sides: x² - 9x + 81/4 = -15 + 81/4 Step 6: Write the left side as a perfect square: (x - 9/2)² = -15 + 81/4 Step 7: Combine the right side: -15 = -60/4, so -60/4 + 81/4 = 21/4 Step 8: The equation is (x - 9/2)² = 21/4 Step 9: In the form (x + h)² = k, we have h = -9/2 and k = 21/4

5. Charlotte is designing a parabolic fountain for a city plaza. The height of the water stream (in meters) above the basin is modeled by h(x) = -4x² + 48x - 80, where x is the horizontal distance (in meters) from the nozzle. 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 water stream and the horizontal distance from the nozzle where this maximum occurs. Then, state the coordinates of the vertex. Answer: Maximum height of 64 meters at a horizontal distance of 6 meters; vertex at (6, 64) Solution: Start with h(x) = -4x² + 48x - 80 Factor out -4 from the x² and x terms: h(x) = -4(x² - 12x) - 80 Complete the square inside the parentheses. Take half of -12, which is -6, and square it to get 36.
   Full step-by-step solution

   Step 1: Start with h(x) = -4x² + 48x - 80 Step 2: Factor out -4 from the x² and x terms: h(x) = -4(x² - 12x) - 80 Step 3: Complete the square inside the parentheses. Take half of -12, which is -6, and square it to get 36. Step 4: Add and subtract 36 inside the parentheses: h(x) = -4(x² - 12x + 36 - 36) - 80 Step 5: Rewrite as: h(x) = -4[(x² - 12x + 36) - 36] - 80 Step 6: Factor the perfect square trinomial: h(x) = -4[(x - 6)² - 36] - 80 Step 7: Distribute the -4: h(x) = -4(x - 6)² + 144 - 80 Step 8: Simplify: h(x) = -4(x - 6)² + 64 Step 9: The vertex form is h(x) = -4(x - 6)² + 64, so the vertex is at (6, 64). Step 10: Since -4 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the water stream is 64 meters, occurring at a horizontal distance of 6 meters from the nozzle. The vertex is (6, 64).

6. A parabolic satellite dish is designed with a cross-section that can be modeled by the equation y = 2x^2 + 12x + 7, where y represents the height (in meters) above the ground and x represents the horizontal distance (in meters) from the left edge of the dish. By completing the square, rewrite this equation in vertex form y = a(x - h)^2 + k to determine the coordinates of the vertex, which represents the lowest point of the dish. Answer: (-3, -11) Solution: Start with the equation y = 2x^2 + 12x + 7. Factor out the coefficient of x^2 from the x terms: y = 2(x^2 + 6x) + 7. Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with the equation y = 2x^2 + 12x + 7. Step 2: Factor out the coefficient of x^2 from the x terms: y = 2(x^2 + 6x) + 7. Step 3: Complete the square inside the parentheses. Take half of 6, which is 3, and square it to get 9. Step 4: Add and subtract 9 inside the parentheses: y = 2(x^2 + 6x + 9 - 9) + 7. Step 5: Rewrite the perfect square trinomial: y = 2[(x + 3)^2 - 9] + 7. Step 6: Distribute the 2: y = 2(x + 3)^2 - 18 + 7. Step 7: Combine constants: y = 2(x + 3)^2 - 11. Step 8: The vertex form is y = 2(x + 3)^2 - 11, so the vertex is at (-3, -11). The vertex is at (-3, -11).

7. Complete the square: 2x² - 12x + 7 = 0 → (x + h)² = k, find h and k? Answer: h = -3, k = 11/2 Solution: Start with 2x² - 12x + 7 = 0 Move the constant term to the right side: 2x² - 12x = -7 Factor out the coefficient of x² from the left side: 2(x² - 6x) = -7 Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with 2x² - 12x + 7 = 0 Step 2: Move the constant term to the right side: 2x² - 12x = -7 Step 3: Factor out the coefficient of x² from the left side: 2(x² - 6x) = -7 Step 4: Complete the square inside the parentheses. Take half of -6: -6/2 = -3. Square it: (-3)² = 9. Step 5: Add and subtract 9 inside the parentheses: 2(x² - 6x + 9 - 9) = -7 Step 6: Rewrite as: 2[(x - 3)² - 9] = -7 Step 7: Distribute the 2: 2(x - 3)² - 18 = -7 Step 8: Add 18 to both sides: 2(x - 3)² = 11 Step 9: Divide both sides by 2: (x - 3)² = 11/2 Step 10: In the form (x + h)² = k, we have (x - 3)² = (x + (-3))², so h = -3 and k = 11/2.