Complete the Square

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Answer Key & Explanations

Complete the Square · Grade 9 · Worksheet 3

1. Noah is designing a rectangular fountain for a city plaza. The length of the fountain is 6 meters more than its width. The area of the fountain is 91 square meters. By rewriting the area equation in the form (x + h)² + k, find the width of the fountain in meters. Answer: 7 Solution: Let the width be x meters. Then the length is (x + 6) meters. Area = width × length, so x(x + 6) = 91.
   Full step-by-step solution

   Step 1: Let the width be x meters. Then the length is (x + 6) meters. Step 2: Area = width × length, so x(x + 6) = 91. Step 3: Expand: x² + 6x = 91. Step 4: To complete the square, take half of the coefficient of x: 6/2 = 3. Square it: 3² = 9. Step 5: Add 9 to both sides to keep the equation balanced: x² + 6x + 9 = 91 + 9. Step 6: The left side is a perfect square: (x + 3)² = 100. Step 7: Take the square root of both sides: x + 3 = 10 or x + 3 = -10. Step 8: Solve for x: x = 7 or x = -13. Step 9: Width cannot be negative, so x = 7. The width of the fountain is 7 meters.

2. 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 completing the square to solve for the width. Answer: 8 meters by 12 meters Solution: Let the width = \( w \) meters. The length is 4 meters longer than the width, so length = \( w + 4 \) meters.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width = \( w \) meters. The length is 4 meters longer than the width, so length = \( w + 4 \) meters. --- **Step 2: Write the area equation** Area = length × width \( 96 = (w + 4) \times w \) --- **Step 3: Expand and rearrange** \( 96 = w^2 + 4w \) Bring all terms to one side: \( w^2 + 4w - 96 = 0 \) --- **Step 4: Prepare to complete the square** We have: \( w^2 + 4w - 96 = 0 \) Move constant term to the right: \( w^2 + 4w = 96 \) --- **Step 5: Complete the square** Take the coefficient of \( w \), which is 4, halve it → 2, square it → 4. Add 4 to both sides: \( w^2 + 4w + 4 = 96 + 4 \) \( w^2 + 4w + 4 = 100 \) --- **Step 6: Factor the perfect square trinomial** Left side: \( (w + 2)^2 = 100 \) --- **Step 7: Solve for \( w \)** Take square root of both sides: \( w + 2 = \pm 10 \) So: Case 1: \( w + 2 = 10 \) → \( w = 8 \) Case 2: \( w + 2 = -10 \) → \( w = -12 \) (not valid, width can’t be negative) --- **Step 8: Find length** Width \( w = 8 \) meters Length \( = w + 4 = 8 + 4 = 12 \) meters --- **Final answer:** The garden is 8 meters by 12 meters.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is circumscribed around the triangle such that all three vertices lie on the circle's circumference. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. Find the length of the hypotenuse using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units The diameter of the circle is 10 units, so the radius is 10/2 = 5 units Calculate the area of…
   Full step-by-step solution

   Step 1: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 2: Find the length of the hypotenuse using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units Step 3: The diameter of the circle is 10 units, so the radius is 10/2 = 5 units Step 4: Calculate the area of the circle using the formula A = πr^2 Step 5: A = 3.14 × 5^2 = 3.14 × 25 = 78.5 The answer is 78.5.

4. Kaia is designing a rectangular meditation garden. The length of the garden is 14 meters more than its width. The area of the garden is 207 square meters. Write the expression for the area in the form (w + h)^2 + k, where w is the width of the garden, and then determine the width of the garden. Answer: 9 meters Solution: Let w be the width in meters. Since the length is 14 meters more than the width, the length is w + 14 meters.
   Full step-by-step solution

   Step 1: Let w be the width in meters. Since the length is 14 meters more than the width, the length is w + 14 meters. Step 2: The area of the garden is width times length, so the equation is: w(w + 14) = 207 Step 3: Expand: w^2 + 14w = 207 Step 4: To complete the square, take half of the coefficient of w (which is 14), square it: (14/2)^2 = 7^2 = 49 Step 5: Add and subtract 49 to the left side: w^2 + 14w + 49 - 49 = 207 Step 6: Rewrite as: (w^2 + 14w + 49) - 49 = 207 Step 7: Factor the perfect square: (w + 7)^2 - 49 = 207 Step 8: Add 49 to both sides: (w + 7)^2 = 256 Step 9: Take the square root of both sides: w + 7 = 16 or w + 7 = -16 Step 10: Solve for w: w = 9 or w = -23 Step 11: Since width cannot be negative, w = 9 meters. The width of the garden is 9 meters.

5. A rectangular garden has a length that is 6 meters more than its width. If the area of the garden is 135 square meters, what is the length of the garden in meters? Answer: 15 Solution: Let the width be x meters. Since the length is 6 meters more than the width, the length is (x + 6) meters.
   Full step-by-step solution

   Step 1: Let the width be x meters. Since the length is 6 meters more than the width, the length is (x + 6) meters. Step 2: The area is given as 135 square meters, so we set up the equation: x(x + 6) = 135 Step 3: Expand the equation: x^2 + 6x = 135 Step 4: To complete the square, move the constant to the right side: x^2 + 6x = 135 Step 5: Take half of the coefficient of x (which is 6), square it: (6/2)^2 = 3^2 = 9 Step 6: Add this value to both sides: x^2 + 6x + 9 = 135 + 9 Step 7: Factor the left side as a perfect square: (x + 3)^2 = 144 Step 8: Take the square root of both sides: x + 3 = 12 or x + 3 = -12 Step 9: Solve for x: x = 9 or x = -15 Step 10: Since width cannot be negative, x = 9 meters Step 11: The length is x + 6 = 9 + 6 = 15 meters The answer is 15.

6. Aroha is designing a rectangular fish pond. The length of the pond is 10 meters more than its width. The area of the pond is 144 square meters. Rewrite the quadratic expression for the area in the form (w + h)² + k, where w is the width of the pond. What is the value of h? Answer: 5 Solution: Let w be the width in meters. The length is w + 10. Area = w(w + 10) = w² + 10w.
   Full step-by-step solution

   Step 1: Let w be the width in meters. The length is w + 10. Area = w(w + 10) = w² + 10w. Step 2: To complete the square for w² + 10w, take half of the coefficient of w: 10/2 = 5. Step 3: Square this value: 5² = 25. Step 4: Write the expression as (w² + 10w + 25) - 25. Step 5: Factor the perfect square: (w + 5)² - 25. So the expression is (w + 5)² - 25. Here h = 5. The answer is 5.

7. x² - 10x + 24 = 0 Answer: x = 4, 6 Solution: Start with x² - 10x + 24 = 0 Move the constant term to the right side: x² - 10x = -24 Take half of the coefficient of x: -10 ÷ 2 = -5 Square this result: (-5)² = 25 Add 25 to both sides: x² - 10x + 25 = -24 + 25 Simplify: x² - 10x + 25 = 1 Factor the left side as a perfect square: (x - 5)² = 1…
   Full step-by-step solution

   Step 1: Start with x² - 10x + 24 = 0 Step 2: Move the constant term to the right side: x² - 10x = -24 Step 3: Take half of the coefficient of x: -10 ÷ 2 = -5 Step 4: Square this result: (-5)² = 25 Step 5: Add 25 to both sides: x² - 10x + 25 = -24 + 25 Step 6: Simplify: x² - 10x + 25 = 1 Step 7: Factor the left side as a perfect square: (x - 5)² = 1 Step 8: Take square root of both sides: x - 5 = ±1 Step 9: Solve for x: x = 5 + 1 = 6 or x = 5 - 1 = 4 The solutions are x = 4 and x = 6.

8. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is inscribed within the triangle such that it is tangent to all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: Area = (1/2) × base × height = (1/2) × 8 × 6 = 24 square units Calculate the hypotenuse using the Pythagorean theorem Hypotenuse = sqrt(8² + 6²) = sqrt(64 + 36) = sqrt(100) = 10 units Perimeter = 8 + 6 + 10 = 24 units Use the formula for the radius of an inscribed circle in a triangle r = (2 ×…
   Full step-by-step solution

   Step 1: Calculate the area of the triangle Area = (1/2) × base × height = (1/2) × 8 × 6 = 24 square units Step 2: Calculate the hypotenuse using the Pythagorean theorem Hypotenuse = sqrt(8² + 6²) = sqrt(64 + 36) = sqrt(100) = 10 units Step 3: Calculate the perimeter of the triangle Perimeter = 8 + 6 + 10 = 24 units Step 4: Use the formula for the radius of an inscribed circle in a triangle r = (2 × Area) / Perimeter r = (2 × 24) / 24 = 48 / 24 = 2 units The answer is 2.