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Quadratic Completing Square

Grade 9 · Algebra · Worksheet 1

  1. The function f(x) = x² - 6x + 2 represents the height of a projectile in meters after x seconds. Find the time when the projectile reaches its minimum height by solving the equation x² - 6x + 2 = 0 using the completing the square method. Enter your answer as the exact value of x. Answer: ______________
  2. Sophia is designing a rectangular rooftop garden for an urban sustainability project. The length of the garden is 10 meters more than its width. The total area of the garden is 144 square meters. To determine if the design fits within the rooftop space, Sophia needs to find the exact dimensions. Set up a quadratic equation for the width (w) and solve it by completing the square to find the width and length of the garden. Answer: ______________
  3. Mason is designing a rectangular courtyard for a new community center. The length of the courtyard is 7 meters more than its width. The area of the courtyard is 98 square meters. To determine the exact dimensions, Mason writes an equation and solves it by completing the square. What are the width and length of the courtyard? Answer: ______________
  4. Hana is designing a rectangular fish pond for her backyard. The area of the pond is 48 square meters. The length of the pond is 4 meters more than twice its width. She needs to find the exact dimensions to buy the correct liner. Set up a quadratic equation and solve it by completing the square to find the width and length of the pond. Answer: ______________
  5. Emma is designing a square courtyard. She wants to add a rectangular fountain in the center such that the fountain's length is 5 meters more than its width. The fountain's area is 66 square meters. If the width of the fountain is represented by x, write a quadratic equation for this situation and solve for x by completing the square. Answer: ______________
  6. Emma is designing a rectangular swimming pool for a community center. The pool will have an area of 150 square meters. The length of the pool is 5 meters more than the width. To ensure proper water circulation, the design team needs the exact dimensions. Solve the quadratic equation by completing the square to find the width and length of the pool. Answer: ______________
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Answer Key & Explanations

Quadratic Completing Square · Grade 9 · Worksheet 1

  1. The function f(x) = x² - 6x + 2 represents the height of a projectile in meters after x seconds. Find the time when the projectile reaches its minimum height by solving the equation x² - 6x + 2 = 0 using the completing the square method. Enter your answer as the exact value of x. Answer: 3 - sqrt(7) Solution: Start with the equation: x² - 6x + 2 = 0 Move the constant term to the right side: x² - 6x = -2 Find the value to complete the square: take half of -6, which is -3, and square it to get 9 Add 9 to both sides: x² - 6x + 9 = -2 + 9 Simplify both sides: (x - 3)² = 7 Take the square root of both…
    Full step-by-step solution

    Step 1: Start with the equation: x² - 6x + 2 = 0 Step 2: Move the constant term to the right side: x² - 6x = -2 Step 3: Find the value to complete the square: take half of -6, which is -3, and square it to get 9 Step 4: Add 9 to both sides: x² - 6x + 9 = -2 + 9 Step 5: Simplify both sides: (x - 3)² = 7 Step 6: Take the square root of both sides: x - 3 = ±sqrt(7) Step 7: Solve for x: x = 3 ± sqrt(7) Since we're looking for the time when the projectile reaches minimum height, and the parabola opens upward, the minimum occurs at the vertex, which is x = 3 - sqrt(7). The answer is 3 - sqrt(7).

  2. Sophia is designing a rectangular rooftop garden for an urban sustainability project. The length of the garden is 10 meters more than its width. The total area of the garden is 144 square meters. To determine if the design fits within the rooftop space, Sophia needs to find the exact dimensions. Set up a quadratic equation for the width (w) and solve it by completing the square to find the width and length of the garden. Answer: width = 3√41 - 5 meters, length = 3√41 + 5 meters Solution: Let w = width of the garden in meters. Then the length = w + 10 meters. Area = width × length, so w(w + 10) = 144.
    Full step-by-step solution

    Step 1: Let w = width of the garden in meters. Then the length = w + 10 meters. Step 2: Area = width × length, so w(w + 10) = 144. Step 3: Expand: w² + 10w = 144. Step 4: To complete the square, take half of 10, which is 5, and square it: 5² = 25. Step 5: Add 25 to both sides: w² + 10w + 25 = 144 + 25. Step 6: Simplify: w² + 10w + 25 = 169. Step 7: Factor the left side as a perfect square: (w + 5)² = 169. Step 8: Take the square root of both sides: w + 5 = ±√169 = ±13. Step 9: Solve for w: w = -5 + 13 = 8 or w = -5 - 13 = -18. Step 10: Since width cannot be negative, w = 8 meters. Step 11: Then length = w + 10 = 8 + 10 = 18 meters. The garden has width 8 meters and length 18 meters.

  3. Mason is designing a rectangular courtyard for a new community center. The length of the courtyard is 7 meters more than its width. The area of the courtyard is 98 square meters. To determine the exact dimensions, Mason writes an equation and solves it by completing the square. What are the width and length of the courtyard? Answer: width = 7 meters, length = 14 meters Solution: Let w = width in meters. Then length = w + 7 meters. Area = width × length, so w(w + 7) = 98.
    Full step-by-step solution

    Step 1: Let w = width in meters. Then length = w + 7 meters. Step 2: Area = width × length, so w(w + 7) = 98. Step 3: Expand: w² + 7w = 98. Step 4: To complete the square, take half of 7, which is 7/2, then square it: (7/2)² = 49/4. Step 5: Add 49/4 to both sides: w² + 7w + 49/4 = 98 + 49/4. Step 6: Write 98 as 392/4: w² + 7w + 49/4 = 392/4 + 49/4 = 441/4. Step 7: The left side is a perfect square: (w + 7/2)² = 441/4. Step 8: Take the square root of both sides: w + 7/2 = ± sqrt(441/4) = ± 21/2. Step 9: Solve for w: w = -7/2 ± 21/2. Step 10: w = (-7 + 21)/2 = 14/2 = 7, or w = (-7 - 21)/2 = -28/2 = -14. Step 11: Since width cannot be negative, w = 7 meters. Step 12: Length = w + 7 = 7 + 7 = 14 meters. The width is 7 meters and the length is 14 meters.

  4. Hana is designing a rectangular fish pond for her backyard. The area of the pond is 48 square meters. The length of the pond is 4 meters more than twice its width. She needs to find the exact dimensions to buy the correct liner. Set up a quadratic equation and solve it by completing the square to find the width and length of the pond. Answer: width = 4 meters, length = 12 meters Solution: Let x represent the width in meters. Then the length is 2x + 4 meters. Area = width × length = x(2x + 4) = 48.
    Full step-by-step solution

    Let x represent the width in meters. Then the length is 2x + 4 meters. Area = width × length = x(2x + 4) = 48. Expand: 2x² + 4x = 48. Divide both sides by 2: x² + 2x = 24. Complete the square: take half of 2, which is 1, square it to get 1. Add 1 to both sides: x² + 2x + 1 = 24 + 1. This gives (x + 1)² = 25. Take the square root of both sides: x + 1 = 5 or x + 1 = -5. Solve for x: x = 4 or x = -6. Since width cannot be negative, x = 4. Then length = 2(4) + 4 = 12. The pond is 4 meters wide and 12 meters long.

  5. Emma is designing a square courtyard. She wants to add a rectangular fountain in the center such that the fountain's length is 5 meters more than its width. The fountain's area is 66 square meters. If the width of the fountain is represented by x, write a quadratic equation for this situation and solve for x by completing the square. Answer: 6 Solution: Let x = width of the fountain. Then length = x + 5. Area = width × length = x(x + 5) = 66.
    Full step-by-step solution

    Step 1: Let x = width of the fountain. Then length = x + 5. Step 2: Area = width × length = x(x + 5) = 66. Step 3: Expand: x² + 5x = 66. Step 4: Move constant to the left: x² + 5x - 66 = 0. Step 5: Complete the square: take half of 5, which is 5/2, then square it: (5/2)² = 25/4. Step 6: Add 25/4 to both sides: x² + 5x + 25/4 = 66 + 25/4. Step 7: Convert 66 to fourths: 66 = 264/4. So 264/4 + 25/4 = 289/4. Step 8: Factor left side as a perfect square: (x + 5/2)² = 289/4. Step 9: Take square root of both sides: x + 5/2 = ± sqrt(289/4) = ± 17/2. Step 10: Solve for x: x = -5/2 + 17/2 = 12/2 = 6, or x = -5/2 - 17/2 = -22/2 = -11. Step 11: Since width cannot be negative, x = 6. The answer is 6.

  6. Emma is designing a rectangular swimming pool for a community center. The pool will have an area of 150 square meters. The length of the pool is 5 meters more than the width. To ensure proper water circulation, the design team needs the exact dimensions. Solve the quadratic equation by completing the square to find the width and length of the pool. Answer: width = -5/2 + (5/2) * sqrt(25) meters, length = 5/2 + (5/2) * sqrt(25) meters Solution: Let w = width in meters. Then length = w + 5. Area = w(w + 5) = 150.
    Full step-by-step solution

    Let w = width in meters. Then length = w + 5. Area = w(w + 5) = 150. Expand: w^2 + 5w = 150. Complete the square: take half of 5, which is 5/2, and square it to get (5/2)^2 = 25/4. Add 25/4 to both sides: w^2 + 5w + 25/4 = 150 + 25/4. The left side is (w + 5/2)^2. Simplify right side: 150 = 600/4, so 600/4 + 25/4 = 625/4. So (w + 5/2)^2 = 625/4. Take square root: w + 5/2 = ± sqrt(625/4) = ± 25/2. Since width is positive: w = -5/2 + 25/2 = 20/2 = 10. Then length = 10 + 5 = 15. The width is 10 meters and the length is 15 meters.