Complete the Square

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Answer Key & Explanations

Complete the Square · Grade 9 · Worksheet 1

1. x² + 12x + 32 = (x + ?)² + ? Answer: 6,-4 Solution: Start with x² + 12x + 32. 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 + 32. Step 2: Take half of the coefficient of x: 12 ÷ 2 = 6. Step 3: Square this result: 6² = 36. Step 4: Rewrite the expression by adding and subtracting 36: x² + 12x + 36 - 36 + 32. Step 5: Group the perfect square trinomial: (x² + 12x + 36) + (-36 + 32). Step 6: Factor the trinomial: (x + 6)² + (-4). Step 7: The expression is now (x + 6)² - 4. The answer is 6,-4.

2. x² + 6x + 5 = 0 Answer: x = -1, -5 Solution: x² + 6x + 5 = 0 Identify the coefficients. The equation is in the form ax² + bx + c = 0. Here, a = 1, b = 6, c = 5.
   Full step-by-step solution

   We are solving the quadratic equation: x² + 6x + 5 = 0 Step 1: Identify the coefficients. The equation is in the form ax² + bx + c = 0. Here, a = 1, b = 6, c = 5. Step 2: Factor the quadratic expression. We look for two numbers that multiply to a*c = 1*5 = 5 and add to b = 6. The numbers 1 and 5 multiply to 5, but 1 + 5 = 6 — yes, that works. Step 3: Write the factored form. Since a = 1, we can factor directly: x² + 6x + 5 = (x + 1)(x + 5) = 0. Step 4: Apply the zero-product property. If (x + 1)(x + 5) = 0, then at least one factor must be zero. So: x + 1 = 0 or x + 5 = 0 Step 5: Solve each equation. x + 1 = 0 → x = -1 x + 5 = 0 → x = -5 Step 6: State the solution. The solutions are x = -1 and x = -5.

3. x² + 8x + 10 = (x + ?)² + ? Answer: 4,-6 Solution: Start with x² + 8x + 10. Take half of the coefficient of x: 8 ÷ 2 = 4. Square this result: 4² = 16.
   Full step-by-step solution

   Step 1: Start with x² + 8x + 10. Step 2: Take half of the coefficient of x: 8 ÷ 2 = 4. Step 3: Square this result: 4² = 16. Step 4: Rewrite the expression as (x² + 8x + 16) + 10 - 16. Step 5: Factor the perfect square trinomial: (x + 4)². Step 6: Combine the constants: 10 - 16 = -6. Step 7: The completed square form is (x + 4)² - 6. The answer is 4,-6.

4. A rectangular garden has a length that is 4 meters longer than its width. If the area of the garden is 60 square meters, what are the dimensions of the garden? Write your answer in the form 'width x length'. Answer: 6 x 10 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 of a rectangle = length × width Given area = 60 square meters, so: \[ w \times (w + 4) = 60 \] --- **Step 3: Expand and rearrange** \[ w^2 + 4w = 60 \] Subtract 60 from both sides: \[ w^2 + 4w - 60 = 0 \] --- **Step 4: Solve the quadratic equation** We solve \( w^2 + 4w - 60 = 0 \) by factoring. We need two numbers whose product is \(-60\) and whose sum is \(4\). Those numbers are \(10\) and \(-6\). So: \[ w^2 + 4w - 60 = (w + 10)(w - 6) = 0 \] --- **Step 5: Find possible values of \( w \)** From \((w + 10)(w - 6) = 0\): \( w + 10 = 0 \) → \( w = -10 \) (not possible, width can't be negative) \( w - 6 = 0 \) → \( w = 6 \) So width \( w = 6 \) meters. --- **Step 6: Find length** Length \( l = w + 4 = 6 + 4 = 10 \) meters. --- **Step 7: Final answer in requested format** Dimensions: width × length = \( 6 \times 10 \) --- **Final answer:** 6 x 10

5. Isabella is designing a rectangular patio. The length of the patio is 14 meters more than its width. The area of the patio is 207 square meters. By completing the square, express the relationship in the form (w + h)² = k, where w is the width of the patio in meters. What is the value of k? Answer: 256 Solution: Let w be the width in meters. Then the length is w + 14 meters. Area = length * width = (w + 14) * w = w² + 14w.
   Full step-by-step solution

   Step 1: Let w be the width in meters. Then the length is w + 14 meters. Step 2: Area = length * width = (w + 14) * w = w² + 14w. Step 3: Set the area equal to 207: w² + 14w = 207. Step 4: To complete the square, take half of the coefficient of w, which is 14/2 = 7, and square it: 7² = 49. Step 5: Add 49 to both sides: w² + 14w + 49 = 207 + 49. Step 6: Simplify the right side: 207 + 49 = 256. Step 7: The left side is a perfect square trinomial: (w + 7)² = 256. Therefore, the equation in the form (w + h)² = k is (w + 7)² = 256, so k = 256. The answer is 256.

6. x² + 8x + 13 = ? (complete the square) Answer: (x + 4)² - 3 Solution: Start with the expression: x² + 8x + 13 Focus on the quadratic and linear terms: x² + 8x Take half of the coefficient of x (which is 8): 8 ÷ 2 = 4 Square this result: 4² = 16 Add and subtract 16 within the expression: x² + 8x + 16 - 16 + 13 Group the perfect square trinomial: (x² + 8x + 16) +…
   Full step-by-step solution

   Step 1: Start with the expression: x² + 8x + 13 Step 2: Focus on the quadratic and linear terms: x² + 8x Step 3: Take half of the coefficient of x (which is 8): 8 ÷ 2 = 4 Step 4: Square this result: 4² = 16 Step 5: Add and subtract 16 within the expression: x² + 8x + 16 - 16 + 13 Step 6: Group the perfect square trinomial: (x² + 8x + 16) + (-16 + 13) Step 7: Factor the perfect square: (x + 4)² Step 8: Combine the constants: -16 + 13 = -3 Step 9: Write the final expression: (x + 4)² - 3

7. Emma is planning a rectangular patio. The length of the patio is 10 meters more than its width. The area of the patio is 375 square meters. By completing the square, rewrite the expression for the area in the form (w + a)^2 + b, where w represents the width of the patio in meters. What are the values of a and b? Answer: a = 5, b = -400 Solution: Let w be the width in meters. Then the length is w + 10 meters. The area is w(w + 10) = 375.
   Full step-by-step solution

   Step 1: Let w be the width in meters. Then the length is w + 10 meters. The area is w(w + 10) = 375. So the quadratic expression is w^2 + 10w. Step 2: To complete the square for w^2 + 10w, take half of 10, which is 5, and square it: 5^2 = 25. Step 3: Add and subtract 25: w^2 + 10w + 25 - 25 = (w + 5)^2 - 25. Step 4: The area expression is (w + 5)^2 - 25 = 375, so rearranging gives (w + 5)^2 - 400 = 0. But the question asks for the expression in the form (w + a)^2 + b. From step 3, we have (w + 5)^2 - 25. However, the full expression from the area equation includes the constant 375 on the other side. The problem says "rewrite the expression for the area", meaning the quadratic w^2 + 10w. So w^2 + 10w = (w + 5)^2 - 25. Thus a = 5 and b = -25. Step 5: But checking the area equation: w^2 + 10w - 375 = 0. Completing the square: (w + 5)^2 - 25 - 375 = 0, so (w + 5)^2 - 400 = 0. The expression for the area (w^2 + 10w) is (w + 5)^2 - 25, so a = 5, b = -25. The answer is a = 5, b = -25.

8. Emma is designing a rectangular fountain for a city park. The fountain's length is 8 meters more than its width. The total area of the fountain and a surrounding decorative tile border of uniform width is 165 square meters. If the border width is 1.5 meters, what are the dimensions of the fountain itself? Answer: 7 meters by 15 meters Solution: When solving problems involving borders around rectangular areas, remember that the border adds to both the length and width. If the original width is w and length is l, and the border has width b, then the total dimensions become (w + 2b) and (l + 2b).
   Full step-by-step solution

   When solving problems involving borders around rectangular areas, remember that the border adds to both the length and width. If the original width is w and length is l, and the border has width b, then the total dimensions become (w + 2b) and (l + 2b). The relationship between length and width is often given, allowing you to create a quadratic equation that can be solved to find the original dimensions.

9. A rectangular garden has a length that is 6 meters longer than its width. If the total area of the garden is 91 square meters, what is the width of the garden? Answer: 7 Solution: Let the width of the garden be \( w \) meters. The length is 6 meters longer than the width, so length \( l = w + 6 \). Area of a rectangle = length × width.
   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 6 meters longer than the width, so length \( l = w + 6 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 91 square meters. 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 it by looking for two numbers that multiply to -91 and add to 6. The pairs of factors of 91: (1, 91), (7, 13). Since 13 - 7 = 6, we can use 13 and -7. So: \[ w^2 + 6w - 91 = (w + 13)(w - 7) = 0 \] --- **Step 5: Find possible values of w** \[ w + 13 = 0 \quad \text{or} \quad w - 7 = 0 \] \[ w = -13 \quad \text{or} \quad w = 7 \] --- **Step 6: Interpret the solutions** Width cannot be negative, so \( w = -13 \) is not valid. Thus, the width is \( w = 7 \) meters. --- **Step 7: Verify** Length = \( 7 + 6 = 13 \) meters. Area = \( 13 \times 7 = 91 \) square meters. Correct. --- **Final answer:** 7