Completing the Square

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Answer Key & Explanations

Completing the Square · Grade 10 · Worksheet 1

1. x² + 6x + 5 = 0 → (x + a)² + b = 0, find a and b Answer: a=3, b=-4 Solution: x^2 + 6x + 5 = 0 (x + a)^2 + b = 0 x^2 + 6x + 5 Take the coefficient of x, which is 6.
   Full step-by-step solution

   We start with the equation: x^2 + 6x + 5 = 0 We want to rewrite it in the form: (x + a)^2 + b = 0 --- **Step 1: Understand completing the square** The general method: x^2 + 6x + 5 Take the coefficient of x, which is 6. Divide it by 2: 6/2 = 3 Square it: 3^2 = 9 --- **Step 2: Add and subtract 9 inside the expression** x^2 + 6x + 5 = x^2 + 6x + 9 - 9 + 5 --- **Step 3: Group the perfect square trinomial** x^2 + 6x + 9 is (x + 3)^2 So we have: (x^2 + 6x + 9) + (-9 + 5) = (x + 3)^2 - 4 --- **Step 4: Write in the required form** (x + 3)^2 - 4 = 0 This matches (x + a)^2 + b = 0 So a = 3, b = -4 --- **Step 5: Check** (x + 3)^2 - 4 = x^2 + 6x + 9 - 4 = x^2 + 6x + 5 ✔ --- **Final answer:** a = 3, b = -4

2. 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

3. Complete the square: x² + 11x + 26 = 0 → (x + a)² = b, find a and b? Answer: a = 11/2, b = -7/4 Solution: Start with x² + 11x + 26 = 0. Move the constant term to the right side: x² + 11x = -26. Take half of the coefficient of x (which is 11): 11/2.
   Full step-by-step solution

   Step 1: Start with x² + 11x + 26 = 0. Step 2: Move the constant term to the right side: x² + 11x = -26. Step 3: Take half of the coefficient of x (which is 11): 11/2. Step 4: Square this value: (11/2)² = 121/4. Step 5: Add 121/4 to both sides: x² + 11x + 121/4 = -26 + 121/4. Step 6: The left side is now a perfect square: (x + 11/2)² = -26 + 121/4. Step 7: Simplify the right side: -26 = -104/4, so -104/4 + 121/4 = 17/4. Step 8: Therefore, (x + 11/2)² = 17/4. So a = 11/2 and b = 17/4. The answer is a = 11/2, b = 17/4.

4. Kaia is designing a suspension bridge for a local park. The shape of the main cable between two towers can be modeled by the quadratic function h(x) = 3x^2 - 42x + 156, where h(x) is the height of the cable in meters above the bridge deck and x is the horizontal distance in meters from the left tower. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to find the minimum height of the cable above the deck and the horizontal distance from the left tower where this minimum occurs. Then, state the coordinates of the vertex. Answer: Minimum height of 9 meters at a horizontal distance of 7 meters; vertex at (7, 9) Solution: Start with the function: h(x) = 3x^2 - 42x + 156 Factor out 3 from the x^2 and x terms: h(x) = 3(x^2 - 14x) + 156 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) = 3x^2 - 42x + 156 Step 2: Factor out 3 from the x^2 and x terms: h(x) = 3(x^2 - 14x) + 156 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) = 3(x^2 - 14x + 49 - 49) + 156 Step 5: Rewrite as: h(x) = 3[(x^2 - 14x + 49) - 49] + 156 Step 6: Factor the perfect square trinomial: h(x) = 3[(x - 7)^2 - 49] + 156 Step 7: Distribute the 3: h(x) = 3(x - 7)^2 - 147 + 156 Step 8: Simplify: h(x) = 3(x - 7)^2 + 9 Step 9: The vertex form is h(x) = 3(x - 7)^2 + 9, so the vertex is at (7, 9). Step 10: Since the coefficient of (x - 7)^2 is positive, the parabola opens upward, and the vertex represents the minimum point. Therefore, the minimum height of the cable is 9 meters above the deck, occurring at a horizontal distance of 7 meters from the left tower. The vertex is (7, 9).

5. A rectangular garden is designed with length (x + 8) meters and width (x + 2) meters. When the area is expressed in the form (x + p)² + q, what are the values of p and q? Answer: p = 5, q = -9 Solution: Length = (x + 8) meters Width = (x + 2) meters Area = Length × Width Area = (x + 8)(x + 2) (x + 8)(x + 2) = x(x + 2) + 8(x + 2) = x² + 2x + 8x + 16 = x² + 10x + 16 So Area = x² + 10x + 16 We complete the square for x² + 10x + 16.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Write the area expression** Length = (x + 8) meters Width = (x + 2) meters Area = Length × Width Area = (x + 8)(x + 2) --- **Step 2: Expand the product** (x + 8)(x + 2) = x(x + 2) + 8(x + 2) = x² + 2x + 8x + 16 = x² + 10x + 16 So Area = x² + 10x + 16 --- **Step 3: Write in the form (x + p)² + q** We complete the square for x² + 10x + 16. Take the x² + 10x part: x² + 10x = (x + 5)² - 25 because (x + 5)² = x² + 10x + 25, so subtract 25 to get x² + 10x. --- **Step 4: Substitute back** Area = [ (x + 5)² - 25 ] + 16 = (x + 5)² - 25 + 16 = (x + 5)² - 9 --- **Step 5: Identify p and q** Comparing with (x + p)² + q: p = 5 q = -9 --- **Final answer:** p = 5, q = -9

6. Hana is designing a parabolic water fountain for a new park. The height of the water stream (in meters) above the fountain basin is given by the quadratic function h(x) = -2x^2 + 16x - 14, 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 18 meters at a horizontal distance of 4 meters; vertex at (4, 18) Solution: Start with the function: h(x) = -2x^2 + 16x - 14 Factor out -2 from the x^2 and x terms: h(x) = -2(x^2 - 8x) - 14 Complete the square inside the parentheses. Take half of -8, which is -4, and square it to get 16.
   Full step-by-step solution

   Step 1: Start with the function: h(x) = -2x^2 + 16x - 14 Step 2: Factor out -2 from the x^2 and x terms: h(x) = -2(x^2 - 8x) - 14 Step 3: Complete the square inside the parentheses. Take half of -8, which is -4, and square it to get 16. Step 4: Add and subtract 16 inside the parentheses: h(x) = -2(x^2 - 8x + 16 - 16) - 14 Step 5: Rewrite as: h(x) = -2[(x^2 - 8x + 16) - 16] - 14 Step 6: Factor the perfect square trinomial: h(x) = -2[(x - 4)^2 - 16] - 14 Step 7: Distribute the -2: h(x) = -2(x - 4)^2 + 32 - 14 Step 8: Simplify: h(x) = -2(x - 4)^2 + 18 Step 9: The vertex form is h(x) = -2(x - 4)^2 + 18, so the vertex is at (4, 18). Step 10: Since the coefficient of (x - 4)^2 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the water stream is 18 meters, occurring at a horizontal distance of 4 meters from the nozzle. The vertex is (4, 18).

7. Complete the square: x² - 12x + 35 = 0 → (x + a)² = b, find a and b? Answer: a = -6, b = 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 Square this value: (-6)² = 36 Add 36 to both sides: x² - 12x + 36 = -35 + 36 Simplify the right side: x² - 12x + 36 = 1 Factor the left side as a perfect square: (x…
   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 Step 4: Square this value: (-6)² = 36 Step 5: Add 36 to both sides: x² - 12x + 36 = -35 + 36 Step 6: Simplify the right side: x² - 12x + 36 = 1 Step 7: Factor the left side as a perfect square: (x - 6)² = 1 Step 8: This is in the form (x + a)² = b, where a = -6 and b = 1. The answer is a = -6, b = 1.