Quadratic Completing Square

lessonbunny.com

Answer Key & Explanations

Quadratic Completing Square · Grade 9 · Worksheet 1

1. Noah is constructing a rectangular sandbox for his community playground. The area of the sandbox is 66 square meters. The length of the sandbox is 1 meter more than twice its width. To order the correct amount of lumber for the frame, Noah needs to find the exact dimensions of the sandbox. Set up a quadratic equation for the width x and solve it by completing the square to determine the width and length of the sandbox. Answer: Width = (-1 + sqrt(529/4)) / 2? Let me compute properly: width = (-1 + sqrt(529/4)) / 2? Actually solving: 2x² + x - 66 = 0 => x² + (1/2)x = 33 => (x + 1/4)² = 33 + 1/16 = 529/16 => x + 1/4 = ± sqrt(529)/4 = ± 23/4 => x = (-1 ± 23)/4 => x = 22/4 = 5.5 or x = -24/4 = -6. Width = 5.5 m, Length = 2(5.5) + 1 = 12 m. Solution: Let x represent the width in meters. The length is 1 meter more than twice the width, so length = 2x + 1. Area = width × length = x(2x + 1) = 66.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. The length is 1 meter more than twice the width, so length = 2x + 1. Step 2: Area = width × length = x(2x + 1) = 66. Step 3: Expand: 2x² + x = 66. Step 4: Divide both sides by 2: x² + (1/2)x = 33. Step 5: Complete the square. Take half of 1/2: (1/2) / 2 = 1/4. Square it: (1/4)² = 1/16. Step 6: Add 1/16 to both sides: x² + (1/2)x + 1/16 = 33 + 1/16. Step 7: Left side is a perfect square: (x + 1/4)² = 529/16 (since 33 = 528/16, plus 1/16 = 529/16). Step 8: Take the square root of both sides: x + 1/4 = ± sqrt(529/16) = ± 23/4. Step 9: Solve for x: x = -1/4 ± 23/4. Step 10: Two possible solutions: x = (-1 + 23)/4 = 22/4 = 5.5, or x = (-1 - 23)/4 = -24/4 = -6. Step 11: Width cannot be negative, so width = 5.5 meters. Step 12: Length = 2(5.5) + 1 = 11 + 1 = 12 meters. Answer: Width = 5.5 meters, Length = 12 meters.

2. A square is drawn on a coordinate plane with vertices at (0,0), (x,0), (x,x+6), and (0,x+6). The area of the square is 121 square units. Write a quadratic equation for x and solve it by completing the square. Answer: 5 Solution: The side length of the square is (x+6) - 0 = x+6. Step 2: Area = side squared = (x+6)^2 = 121. Step 3: Expand: x^2 + 12x + 36 = 121.
   Full step-by-step solution

   Step 1: The side length of the square is (x+6) - 0 = x+6. Step 2: Area = side squared = (x+6)^2 = 121. Step 3: Expand: x^2 + 12x + 36 = 121. Step 4: Subtract 121 from both sides: x^2 + 12x + 36 - 121 = 0, so x^2 + 12x - 85 = 0. Step 5: Move constant: x^2 + 12x = 85. Step 6: Complete the square: half of 12 is 6, square it to get 36. Add 36 to both sides: x^2 + 12x + 36 = 85 + 36. Step 7: x^2 + 12x + 36 = 121. Step 8: Factor: (x + 6)^2 = 121. Step 9: Take square root: x + 6 = ±11. Step 10: Solve: x = -6 + 11 = 5, or x = -6 - 11 = -17. Step 11: Since length cannot be negative, x = 5. The answer is 5.

3. x² + 16x + 21 = 0 Answer: -8 ± √43 Solution: Move the constant term to the right side: x² + 16x = -21 Find the number to complete the square: (16/2)² = 64 Add 64 to both sides: x² + 16x + 64 = -21 + 64 Factor the perfect square trinomial: (x + 8)² = 43 Take square root of both sides: x + 8 = ±√43 Solve for x: x = -8 ± √43 The answer is -8…
   Full step-by-step solution

   Step 1: Move the constant term to the right side: x² + 16x = -21 Step 2: Find the number to complete the square: (16/2)² = 64 Step 3: Add 64 to both sides: x² + 16x + 64 = -21 + 64 Step 4: Factor the perfect square trinomial: (x + 8)² = 43 Step 5: Take square root of both sides: x + 8 = ±√43 Step 6: Solve for x: x = -8 ± √43 The answer is -8 ± √43.

4. A rectangular garden has a length that is 4 meters more than its width. If the area of the garden is 96 square meters, find the dimensions of the garden by solving the quadratic equation using the completing the square method. Answer: width = 8 meters, length = 12 meters Solution: Let the width = \( w \) meters. Length is 4 meters more than width, so length = \( w + 4 \) meters.
   Full step-by-step solution

   Let's solve step-by-step using the completing the square method. --- **Step 1: Define variables** Let the width = \( w \) meters. Length is 4 meters more than width, so length = \( w + 4 \) meters. --- **Step 2: Write the area equation** Area = length × width \( 96 = (w + 4) \times w \) \( 96 = w^2 + 4w \) --- **Step 3: Rearrange for completing the square** \( w^2 + 4w = 96 \) --- **Step 4: Complete the square** Take the coefficient of \( w \), which is 4, divide by 2 → \( 2 \), square it → \( 4 \). Add 4 to both sides: \( w^2 + 4w + 4 = 96 + 4 \) \( w^2 + 4w + 4 = 100 \) --- **Step 5: Factor the perfect square trinomial** Left side: \( (w + 2)^2 \) So: \( (w + 2)^2 = 100 \) --- **Step 6: Solve for \( w \)** Take square root of both sides: \( w + 2 = 10 \) or \( w + 2 = -10 \) Case 1: \( w + 2 = 10 \) → \( w = 8 \) Case 2: \( w + 2 = -10 \) → \( w = -12 \) (not valid, width can't be negative) So \( w = 8 \) meters. --- **Step 7: Find length** Length = \( w + 4 = 8 + 4 = 12 \) meters. --- **Final answer:** Width = 8 meters, Length = 12 meters.

5. Noah is designing a rectangular koi pond for a community park. The area of the pond is 126 square meters. The length of the pond is 6 meters more than its width. To determine the exact dimensions for the construction plans, Noah needs to solve the quadratic equation by completing the square. Find the width and length of the pond. Answer: width = -3 + 3√15 meters, length = 3 + 3√15 meters Solution: Let w = width in meters. Then length = w + 6 meters. Area = width × length = w(w + 6) = 126 Expand: w² + 6w = 126 Complete the square.
   Full step-by-step solution

   Step 1: Let w = width in meters. Then length = w + 6 meters. Step 2: Area = width × length = w(w + 6) = 126 Step 3: Expand: w² + 6w = 126 Step 4: Complete the square. Take half of 6, which is 3, and square it: 3² = 9. Step 5: Add 9 to both sides: w² + 6w + 9 = 126 + 9 Step 6: Simplify: w² + 6w + 9 = 135 Step 7: Factor left side: (w + 3)² = 135 Step 8: Take square root of both sides: w + 3 = ±√135 Step 9: Simplify √135 = √(9 × 15) = 3√15 Step 10: So w + 3 = ±3√15 Step 11: Solve for w: w = -3 ± 3√15 Step 12: Only the positive width makes sense: w = -3 + 3√15 meters Step 13: Length = w + 6 = (-3 + 3√15) + 6 = 3 + 3√15 meters The width is -3 + 3√15 meters and the length is 3 + 3√15 meters.

6. Zoe is throwing a ball straight up from the ground. The height of the ball after t seconds is given by h(t) = -16t² + 64t + 21. Using completing the square, find the time in seconds when the ball reaches its maximum height. Answer: 2 Solution: The height function is h(t) = -16t² + 64t + 21 Factor out -16: h(t) = -16(t² - (64/16)t) + 21 Complete the square inside: t² - (64/16)t + (64/32)² - (64/32)² h(t) = -16[(t - 64/32)² - (64/32)²] + 21 h(t) = -16(t - 64/32)² + 16(64/32)² + 21 h(t) = -16(t - 64/32)² + 64²/64 + 21 Maximum height…
   Full step-by-step solution

   Step 1: The height function is h(t) = -16t² + 64t + 21 Step 2: Factor out -16: h(t) = -16(t² - (64/16)t) + 21 Step 3: Complete the square inside: t² - (64/16)t + (64/32)² - (64/32)² Step 4: h(t) = -16[(t - 64/32)² - (64/32)²] + 21 Step 5: h(t) = -16(t - 64/32)² + 16(64/32)² + 21 Step 6: h(t) = -16(t - 64/32)² + 64²/64 + 21 Step 7: Maximum height occurs when t = 64/32 seconds Step 8: Therefore the time is 2 seconds.

7. 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.