Complete the Square

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Answer Key & Explanations

Complete the Square · Grade 9 · Worksheet 1

1. x² + 12x + 29 = (x + ?)² + ? Answer: 6,-7 Solution: Start with x² + 12x + 29. Take half of the coefficient of x: 12 ÷ 2 = 6. Square this result: 6² = 36.
   Full step-by-step solution

   Step 1: Start with x² + 12x + 29. Step 2: Take half of the coefficient of x: 12 ÷ 2 = 6. Step 3: Square this result: 6² = 36. Step 4: Rewrite the expression as (x² + 12x + 36) + 29 - 36. Step 5: Factor the perfect square trinomial: (x + 6)². Step 6: Combine the constants: 29 - 36 = -7. Step 7: The completed square form is (x + 6)² - 7. The answer is 6,-7.

2. x² - 6x + 5 = (x - ?)² - 4 Answer: 3 Solution: Start with the original expression: x² - 6x + 5 To complete the square, take half of the coefficient of x: -6/2 = -3 Square this value: (-3)² = 9 Add and subtract 9: x² - 6x + 9 - 9 + 5 Group the perfect square trinomial: (x² - 6x + 9) + (-9 + 5) Write as a squared binomial: (x - 3)² + (-4)…
   Full step-by-step solution

   Step 1: Start with the original expression: x² - 6x + 5 Step 2: To complete the square, take half of the coefficient of x: -6/2 = -3 Step 3: Square this value: (-3)² = 9 Step 4: Add and subtract 9: x² - 6x + 9 - 9 + 5 Step 5: Group the perfect square trinomial: (x² - 6x + 9) + (-9 + 5) Step 6: Write as a squared binomial: (x - 3)² + (-4) Step 7: Compare with the given form: (x - 3)² - 4 Step 8: The missing value is 3 The answer is 3.

3. A rectangular garden has a length that is 2 meters more than its width. If the area of the garden is 120 square meters, find the dimensions of the garden by completing the square. Answer: 10 x 12 Solution: Let the width be x meters. Since the length is 2 meters more than the width, the length is (x + 2) meters.
   Full step-by-step solution

   Step 1: Let the width be x meters. Since the length is 2 meters more than the width, the length is (x + 2) meters. Step 2: The area is length × width = x(x + 2) = 120 Step 3: Expand the equation: x² + 2x = 120 Step 4: To complete the square, take half of the coefficient of x (which is 2) and square it: (2/2)² = 1² = 1 Step 5: Add this value to both sides: x² + 2x + 1 = 120 + 1 Step 6: Factor the left side: (x + 1)² = 121 Step 7: Take the square root of both sides: x + 1 = 11 or x + 1 = -11 Step 8: Solve for x: x = 10 or x = -12 Step 9: Since width cannot be negative, x = 10 meters Step 10: The length is x + 2 = 12 meters Therefore, the garden dimensions are 10 meters by 12 meters.

4. Emma is analyzing a rectangular garden with length (x + 10) meters and width (x + 5) meters. She draws a square inside the garden that has side length (x + 5) meters, leaving a border of uniform width around it. The total area of the garden is given by the expression x² + 15x + 50. Rewrite this quadratic expression in the form (x + h)² + k by completing the square, and state the values of h and k. Answer: (x + 7.5)² - 6.25; h = 7.5, k = -6.25 Solution: Start with the expression x² + 15x + 50. To complete the square, focus on x² + 15x. Take half of the coefficient of x: half of 15 is 7.5.
   Full step-by-step solution

   Step 1: Start with the expression x² + 15x + 50. Step 2: To complete the square, focus on x² + 15x. Take half of the coefficient of x: half of 15 is 7.5. Square it: (7.5)² = 56.25. Step 3: Rewrite x² + 15x as (x + 7.5)² - 56.25. Step 4: Add the constant term: (x + 7.5)² - 56.25 + 50 = (x + 7.5)² - 6.25. Step 5: Therefore, h = 7.5 and k = -6.25. The expression in completed square form is (x + 7.5)² - 6.25.

5. Sophia is designing a rectangular flower bed for a community park. The length of the flower bed is 11 meters more than its width. The area of the flower bed is 152 square meters. By completing the square, rewrite the area equation in the form (w + a)² = b, where w represents the width in meters. What is the value of b? Answer: 182.25 Solution: Let w be the width in meters. Since the length is 11 meters more than the width, length = w + 11. Area = width × length = w(w + 11) = 152.
   Full step-by-step solution

   Step 1: Let w be the width in meters. Since the length is 11 meters more than the width, length = w + 11. Step 2: Area = width × length = w(w + 11) = 152. Step 3: Expand: w² + 11w = 152. Step 4: To complete the square, take half of the coefficient of w (which is 11): 11/2 = 5.5. Square it: (5.5)² = 30.25. Step 5: Add 30.25 to both sides: w² + 11w + 30.25 = 152 + 30.25 = 182.25. Step 6: The left side is a perfect square: (w + 5.5)² = 182.25. The value of b is 182.25.

6. x² + 14x + 41 = (x + ?)² + ? Answer: 7,-8 Solution: Start with x² + 14x + 41. Take half of the coefficient of x: 14 ÷ 2 = 7. Square this result: 7² = 49.
   Full step-by-step solution

   Step 1: Start with x² + 14x + 41. Step 2: Take half of the coefficient of x: 14 ÷ 2 = 7. Step 3: Square this result: 7² = 49. Step 4: Rewrite the expression as (x² + 14x + 49) + 41 - 49. Step 5: Factor the perfect square trinomial: (x + 7)². Step 6: Combine the constants: 41 - 49 = -8. Step 7: The completed square form is (x + 7)² - 8. The answer is 7,-8.

7. A rectangular garden has a length that is 6 feet more than its width. If the area of the garden is 91 square feet, what is the width of the garden in feet? Answer: 7 Solution: Let the width of the garden be \( w \) feet. The length is 6 feet more than the width, so length \( l = w + 6 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) feet. The length is 6 feet more than the width, so length \( l = w + 6 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 91 square feet. So: \[ w \times (w + 6) = 91 \] --- **Step 3: Expand and rearrange** \[ w^2 + 6w = 91 \] \[ w^2 + 6w - 91 = 0 \] --- **Step 4: Solve the quadratic equation** We can factor: We need two numbers whose product is \(-91\) and whose sum is \(6\). Try \(13\) and \(-7\): \(13 \times (-7) = -91\) \(13 + (-7) = 6\) ✓ So: \[ (w + 13)(w - 7) = 0 \] --- **Step 5: Find possible values of \( w \)** \( w + 13 = 0 \) → \( w = -13 \) (not possible, width can't be negative) \( w - 7 = 0 \) → \( w = 7 \) --- **Step 6: Verify** Width \( w = 7 \) feet Length \( l = 7 + 6 = 13 \) feet Area = \( 7 \times 13 = 91 \) square feet ✓ --- **Final answer:** The width is 7 feet.

8. A rectangular garden has a length that is 4 meters more than its width. If the area of the garden is 96 square meters, what is the length of the garden in meters? Answer: 12 Solution: Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \). Area of a rectangle = length × width Given area = 96 square meters.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width Given area = 96 square meters. So: \[ l \times w = 96 \] Substitute \( l = w + 4 \): \[ (w + 4) \times w = 96 \] --- **Step 3: Expand and rearrange** \[ w^2 + 4w = 96 \] \[ w^2 + 4w - 96 = 0 \] --- **Step 4: Solve the quadratic equation** We can factor: \[ w^2 + 4w - 96 = 0 \] Look for two numbers whose product is -96 and sum is 4. These numbers are 12 and -8. So: \[ (w + 12)(w - 8) = 0 \] --- **Step 5: Find possible values for w** \[ w + 12 = 0 \quad \text{or} \quad w - 8 = 0 \] \[ w = -12 \quad \text{or} \quad w = 8 \] Width cannot be negative, so \( w = 8 \) meters. --- **Step 6: Find length** \[ l = w + 4 = 8 + 4 = 12 \ \text{meters} \] --- **Step 7: Check** Area = length × width = 12 × 8 = 96, which matches the problem. --- **Final answer:** The length is 12 meters.

9. The quadratic expression 9x² - 42x + k can be written in the form a(x - h)². What is the value of k that makes this a perfect square trinomial? Answer: 49 Solution: For a quadratic expression ax² + bx + c to be a perfect square trinomial, it must satisfy the relationship c = (b/(2a))². Calculate b/(2a) = -42/(2×9) = -42/18 = -7/3. Square this result: (-7/3)² = 49/9.
   Full step-by-step solution

   Step 1: For a quadratic expression ax² + bx + c to be a perfect square trinomial, it must satisfy the relationship c = (b/(2a))². Step 2: In this problem, a = 9 and b = -42. Step 3: Calculate b/(2a) = -42/(2×9) = -42/18 = -7/3. Step 4: Square this result: (-7/3)² = 49/9. Step 5: Since the expression is a(x - h)², the constant term k = a × (49/9) = 9 × (49/9) = 49. Step 6: Therefore, k = 49 makes 9x² - 42x + 49 a perfect square trinomial. The answer is 49.