Quadratic Completing Square

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Answer Key & Explanations

Quadratic Completing Square · Grade 9 · Worksheet 3

1. A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 40 square meters, find the dimensions of the garden by setting up a quadratic equation and solving it by completing the square. Answer: width = 5 meters, length = 8 meters Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than the width, so length \( l = w + 3 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than the width, so length \( l = w + 3 \). --- **Step 2: Write the area equation** Area = length × width \( 40 = (w + 3) \times w \) So: \( w(w + 3) = 40 \) --- **Step 3: Expand and rearrange into standard quadratic form** \( w^2 + 3w = 40 \) \( w^2 + 3w - 40 = 0 \) --- **Step 4: Solve by completing the square** We have: \( w^2 + 3w - 40 = 0 \) First, move the constant term to the right side: \( w^2 + 3w = 40 \) --- **Step 5: Complete the square** Take the coefficient of \( w \), which is 3, halve it: \( 3/2 \), square it: \( (3/2)^2 = 9/4 \). Add \( 9/4 \) to both sides: \( w^2 + 3w + 9/4 = 40 + 9/4 \) --- **Step 6: Simplify the right-hand side** \( 40 + 9/4 = 160/4 + 9/4 = 169/4 \) So: \( w^2 + 3w + 9/4 = 169/4 \) --- **Step 7: Write the left-hand side as a square** \( (w + 3/2)^2 = 169/4 \) --- **Step 8: Take the square root of both sides** \( w + 3/2 = \pm \sqrt{169/4} \) \( w + 3/2 = \pm 13/2 \) --- **Step 9: Solve for \( w \)** Case 1: \( w + 3/2 = 13/2 \) \( w = 13/2 - 3/2 = 10/2 = 5 \) Case 2: \( w + 3/2 = -13/2 \) \( w = -13/2 - 3/2 = -16/2 = -8 \) (not valid since width can't be negative) --- **Step 10: Find length** Width \( w = 5 \) meters Length \( l = w + 3 = 5 + 3 = 8 \) meters --- **Final answer:** width = 5 meters, length = 8 meters

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. 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.

4. 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.

5. Aroha is designing a rectangular community garden. The length of the garden is 9 meters more than its width. The area of the garden is 112 square meters. Aroha wants to know the exact width and length of the garden so she can order fencing. Use completing the square to find the width and length of the garden. Answer: width = sqrt(193)/2 - 9/2 meters, length = sqrt(193)/2 + 9/2 meters Solution: Let x represent the width in meters. Then the length is x + 9 meters.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. Then the length is x + 9 meters. Step 2: Area = width × length = x(x + 9) = 112 Step 3: Expand: x² + 9x = 112 Step 4: To complete the square, take half of 9, which is 9/2, and square it: (9/2)² = 81/4 Step 5: Add 81/4 to both sides: x² + 9x + 81/4 = 112 + 81/4 Step 6: Convert 112 to fourths: 112 = 448/4, so 448/4 + 81/4 = 529/4 Step 7: The left side is a perfect square: (x + 9/2)² = 529/4 Step 8: Take the square root of both sides: x + 9/2 = ± sqrt(529/4) = ± sqrt(529)/2 = ± 23/2 Step 9: Solve for x: x = -9/2 ± 23/2 Step 10: Two possible solutions: x = (-9 + 23)/2 = 14/2 = 7, or x = (-9 - 23)/2 = -32/2 = -16 Step 11: Since width cannot be negative, x = 7 meters. Step 12: Length = x + 9 = 7 + 9 = 16 meters. Step 13: The width is 7 meters and the length is 16 meters. Answer: width = 7 meters, length = 16 meters

6. Olivia is designing a rectangular koi pond for her backyard. The pond will have an area of 135 square meters. She wants the length to be 6 meters more than the width. To order the correct liner, she needs the exact dimensions of the pond. Set up a quadratic equation for this situation and solve it by completing the square to find the width and length of the pond. Answer: width = -3 + 3√15 meters, length = 3 + 3√15 meters Solution: Let x represent the width in meters. Then the length is x + 6 meters.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. Then the length is x + 6 meters. Step 2: Area = width × length = x(x + 6) = 135 Step 3: Expand: x² + 6x = 135 Step 4: Move the constant to the right side: x² + 6x = 135 Step 5: Complete the square: take half of 6, which is 3, then square it to get 9. Add 9 to both sides: x² + 6x + 9 = 135 + 9 Step 6: The left side is now a perfect square trinomial: (x + 3)² = 144 Step 7: Take the square root of both sides: x + 3 = ±√144 Step 8: Simplify: x + 3 = ±12 Step 9: Solve for x: x + 3 = 12 → x = 9 x + 3 = -12 → x = -15 (reject, width cannot be negative) Step 10: So width = 9 meters, length = 9 + 6 = 15 meters. Final answer: width = 9 meters, length = 15 meters.

7. A rectangular garden has a length that is 4 meters longer than its width. If the area of the garden is 96 square meters, find the dimensions of the garden by setting up a quadratic equation and solving by completing the square. Answer: width = 8 meters, length = 12 meters Solution: Let the width of the garden be \( w \) meters. The length is 4 meters longer than the width, so length \( l = w + 4 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 4 meters longer than the width, so length \( l = w + 4 \). --- **Step 2: Write the area equation** Area = length × width \( 96 = (w + 4) \times w \) So: \[ w(w + 4) = 96 \] \[ w^2 + 4w = 96 \] --- **Step 3: Set up for completing the square** We have: \[ w^2 + 4w = 96 \] To complete the square, take half of the coefficient of \( w \), square it, and add to both sides. Coefficient of \( w \) is 4. Half of 4 is 2. Square of 2 is 4. Add 4 to both sides: \[ w^2 + 4w + 4 = 96 + 4 \] \[ w^2 + 4w + 4 = 100 \] --- **Step 4: Factor the left side** The left side is a perfect square trinomial: \[ (w + 2)^2 = 100 \] --- **Step 5: Solve for \( w \)** Take square root of both sides: \[ w + 2 = \pm 10 \] So: Case 1: \( w + 2 = 10 \) → \( w = 8 \) Case 2: \( w + 2 = -10 \) → \( w = -12 \) Since width can't be negative, \( w = 8 \). --- **Step 6: Find length** Length \( l = w + 4 = 8 + 4 = 12 \). --- **Step 7: Check** Area = \( 12 \times 8 = 96 \) square meters. Correct. --- **Final answer:** Width = 8 meters, Length = 12 meters