Completing the Square

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Answer Key & Explanations

Completing the Square · Grade 10 · Worksheet 3

1. Sophia is designing a parabolic arch for a new greenhouse entrance. The height of the arch (in meters) above the ground is given by the quadratic function h(x) = -4x^2 + 48x - 108, 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 36 meters at a horizontal distance of 6 meters; vertex at (6, 36) Solution: Start with the function: h(x) = -4x^2 + 48x - 108 Factor out -4 from the x^2 and x terms: h(x) = -4(x^2 - 12x) - 108 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 the function: h(x) = -4x^2 + 48x - 108 Step 2: Factor out -4 from the x^2 and x terms: h(x) = -4(x^2 - 12x) - 108 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(x^2 - 12x + 36 - 36) - 108 Step 5: Rewrite as: h(x) = -4[(x^2 - 12x + 36) - 36] - 108 Step 6: Factor the perfect square trinomial: h(x) = -4[(x - 6)^2 - 36] - 108 Step 7: Distribute the -4: h(x) = -4(x - 6)^2 + 144 - 108 Step 8: Simplify: h(x) = -4(x - 6)^2 + 36 Step 9: The vertex form is h(x) = -4(x - 6)^2 + 36, so the vertex is at (6, 36). Step 10: Since the coefficient of (x - 6)^2 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the arch is 36 meters, occurring at a horizontal distance of 6 meters from the left base. The vertex is (6, 36).

2. Tane is designing a parabolic skateboard half-pipe for a community park. The cross-section of the ramp can be modeled by the quadratic function h(x) = -5x^2 + 50x - 45, where h(x) is the height in centimeters above the ground and x is the horizontal distance in centimeters from the left edge of the ramp. 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 ramp and the horizontal distance from the left edge where this maximum occurs. Then, state the coordinates of the vertex. Answer: Maximum height of 80 cm at a horizontal distance of 5 cm; vertex at (5, 80) Solution: Start with the function: h(x) = -5x^2 + 50x - 45 Factor out -5 from the x^2 and x terms: h(x) = -5(x^2 - 10x) - 45 Complete the square inside the parentheses. Take half of -10, which is -5, and square it to get 25.
   Full step-by-step solution

   Step 1: Start with the function: h(x) = -5x^2 + 50x - 45 Step 2: Factor out -5 from the x^2 and x terms: h(x) = -5(x^2 - 10x) - 45 Step 3: Complete the square inside the parentheses. Take half of -10, which is -5, and square it to get 25. Step 4: Add and subtract 25 inside the parentheses: h(x) = -5(x^2 - 10x + 25 - 25) - 45 Step 5: Rewrite as: h(x) = -5[(x^2 - 10x + 25) - 25] - 45 Step 6: Factor the perfect square trinomial: h(x) = -5[(x - 5)^2 - 25] - 45 Step 7: Distribute the -5: h(x) = -5(x - 5)^2 + 125 - 45 Step 8: Simplify: h(x) = -5(x - 5)^2 + 80 Step 9: The vertex form is h(x) = -5(x - 5)^2 + 80, so the vertex is at (5, 80). Step 10: Since the coefficient of (x - 5)^2 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum height of the ramp is 80 centimeters, occurring at a horizontal distance of 5 centimeters from the left edge. The vertex is (5, 80).

3. Complete the square: x² + 14x + 29 = 0 → (x + a)² = b, find a and b? Answer: a=7, b=20 Solution: Start with x² + 14x + 29 = 0 Move the constant term to the right side: x² + 14x = -29 Take half of the coefficient of x (14/2 = 7) and square it: 7² = 49 Add 49 to both sides: x² + 14x + 49 = -29 + 49 The left side is a perfect square: (x + 7)² = 20 Therefore, a = 7 and b = 20.
   Full step-by-step solution

   Step 1: Start with x² + 14x + 29 = 0 Step 2: Move the constant term to the right side: x² + 14x = -29 Step 3: Take half of the coefficient of x (14/2 = 7) and square it: 7² = 49 Step 4: Add 49 to both sides: x² + 14x + 49 = -29 + 49 Step 5: The left side is a perfect square: (x + 7)² = 20 Step 6: Therefore, a = 7 and b = 20.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,6). The hypotenuse has length 10 units. By completing the square, determine the value of x. Answer: 8 Solution: Step 1: Apply the Pythagorean theorem to the right triangle with legs x and 6, and hypotenuse 10: x^2 + 6^2 = 10^2 Step 2: Simplify: x^2 + 36 = 100 Step 3: Rearrange to standard quadratic form: x^2 - 64 = 0 Step 4: Complete the square: x^2 + 0x - 64 = 0 Step 5: Move constant term: x^2 + 0x = 64…
   Full step-by-step solution

   Step 1: Apply the Pythagorean theorem to the right triangle with legs x and 6, and hypotenuse 10: x^2 + 6^2 = 10^2 Step 2: Simplify: x^2 + 36 = 100 Step 3: Rearrange to standard quadratic form: x^2 - 64 = 0 Step 4: Complete the square: x^2 + 0x - 64 = 0 Step 5: Move constant term: x^2 + 0x = 64 Step 6: Add (b/2)^2 to both sides: x^2 + 0x + (0/2)^2 = 64 + 0 Step 7: Factor perfect square trinomial: (x + 0)^2 = 64 Step 8: Take square root of both sides: x + 0 = ±8 Step 9: Since length must be positive: x = 8 The answer is 8.

5. A rectangular garden has a length that is 6 meters longer than its width. If the area of the garden is 91 square meters, find the dimensions of the garden by completing the square. Answer: 7 meters by 13 meters Solution: Let the width of the garden be w meters. The length is 6 meters longer than the width, so length = w + 6 meters. The area is length × width = (w + 6) × w = 91.
   Full step-by-step solution

   Let the width of the garden be w meters. The length is 6 meters longer than the width, so length = w + 6 meters. The area is length × width = (w + 6) × w = 91. So: w(w + 6) = 91 w^2 + 6w = 91 We will complete the square. Step 1: Move constant term to the right side. w^2 + 6w = 91 Step 2: Find the number to complete the square. Take half of the coefficient of w: half of 6 is 3. Square it: 3^2 = 9. Step 3: Add 9 to both sides. w^2 + 6w + 9 = 91 + 9 w^2 + 6w + 9 = 100 Step 4: Write the left side as a perfect square. (w + 3)^2 = 100 Step 5: Take the square root of both sides. w + 3 = ±√100 w + 3 = ±10 Step 6: Solve for w. Case 1: w + 3 = 10 → w = 7 Case 2: w + 3 = -10 → w = -13 Since width cannot be negative, w = 7 meters. Length = w + 6 = 7 + 6 = 13 meters. Thus, the dimensions are 7 meters by 13 meters.

6. Mere is designing a parabolic water feature for a community garden. The height of the water stream (in meters) above the basin is given by the quadratic function h(x) = -3x^2 + 24x - 33, 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 15 meters at a horizontal distance of 4 meters; vertex at (4, 15) Solution: Start with the function: h(x) = -3x^2 + 24x - 33 Factor out -3 from the x^2 and x terms: h(x) = -3(x^2 - 8x) - 33 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) = -3x^2 + 24x - 33 Step 2: Factor out -3 from the x^2 and x terms: h(x) = -3(x^2 - 8x) - 33 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) = -3(x^2 - 8x + 16 - 16) - 33 Step 5: Rewrite as: h(x) = -3[(x^2 - 8x + 16) - 16] - 33 Step 6: Factor the perfect square trinomial: h(x) = -3[(x - 4)^2 - 16] - 33 Step 7: Distribute the -3: h(x) = -3(x - 4)^2 + 48 - 33 Step 8: Simplify: h(x) = -3(x - 4)^2 + 15 Step 9: The vertex form is h(x) = -3(x - 4)^2 + 15, so the vertex is at (4, 15). 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 15 meters, occurring at a horizontal distance of 4 meters from the nozzle. The vertex is (4, 15).