Complete the Square

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Complete the Square · Grade 9 · Worksheet 3

1. 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.

2. Emma is analyzing the graph of the quadratic function f(x) = x² + 10x + 15. She notices that the parabola can be rewritten in vertex form by completing the square. What is the vertex of the parabola? Answer: (-5, -10) Solution: Start with f(x) = x² + 10x + 15. To complete the square, take half of the coefficient of x: 10/2 = 5. Square this value: 5² = 25.
   Full step-by-step solution

   Step 1: Start with f(x) = x² + 10x + 15. Step 2: To complete the square, take half of the coefficient of x: 10/2 = 5. Step 3: Square this value: 5² = 25. Step 4: Add and subtract 25 inside the expression: f(x) = x² + 10x + 25 - 25 + 15. Step 5: Group the perfect square trinomial: f(x) = (x² + 10x + 25) + (-25 + 15). Step 6: Factor the trinomial: (x² + 10x + 25) = (x + 5)². Step 7: Simplify the constant: -25 + 15 = -10. Step 8: So f(x) = (x + 5)² - 10. Step 9: The vertex form is f(x) = (x - h)² + k, where the vertex is (h, k). Here, (x + 5) = (x - (-5)), so h = -5 and k = -10. The vertex is (-5, -10).

3. The quadratic expression x² + 14x + k is a perfect square trinomial. What is the value of k? Answer: 49 Solution: A perfect square trinomial has the form (x + a)² = x² + 2ax + a². We are given: x² + 14x + k Compare the given expression to the perfect square form.
   Full step-by-step solution

   A perfect square trinomial has the form (x + a)² = x² + 2ax + a². We are given: x² + 14x + k Step 1: Compare the given expression to the perfect square form. The given expression is: x² + 14x + k The perfect square form is: x² + 2ax + a² Step 2: Match the coefficients of the x terms. From the given expression, the coefficient of x is 14. From the perfect square form, the coefficient of x is 2a. So we set them equal: 2a = 14. Step 3: Solve for 'a'. Divide both sides by 2: a = 14/2 = 7. Step 4: Find the value of k. In the perfect square form, the constant term is a². We found a = 7, so a² = 7² = 49. Therefore, k must be 49. Step 5: Verify the result. If k = 49, the expression becomes x² + 14x + 49. This factors as (x + 7)(x + 7) = (x + 7)², which is indeed a perfect square trinomial. Final answer: k = 49.

4. Noah is designing a rectangular fish pond for a community garden. The length of the pond is 9 meters more than its width. The area of the pond is 136 square meters. By completing the square, express the relationship between the width and area in the form (x + h)^2 + k = 0, where x represents the width of the pond. Answer: (x + 4.5)^2 - 156.25 = 0 Solution: Let x be the width of the pond in meters. Since the length is 9 meters more than the width, the length is (x + 9) meters. Area = width * length = x(x + 9) = 136.
   Full step-by-step solution

   Step 1: Let x be the width of the pond in meters. Since the length is 9 meters more than the width, the length is (x + 9) meters. Step 2: Area = width * length = x(x + 9) = 136. Step 3: Expand: x^2 + 9x = 136. Step 4: Bring the constant to the left side: x^2 + 9x - 136 = 0. Step 5: To complete the square, take half of the coefficient of x: 9/2 = 4.5. Square it: (4.5)^2 = 20.25. Step 6: Add and subtract 20.25 within the expression: x^2 + 9x + 20.25 - 20.25 - 136 = 0. Step 7: Rewrite as a perfect square: (x + 4.5)^2 - 20.25 - 136 = 0. Step 8: Combine constants: -20.25 - 136 = -156.25. Step 9: So the expression is (x + 4.5)^2 - 156.25 = 0. The answer is (x + 4.5)^2 - 156.25 = 0.

5. A rectangular garden has a length that is 6 meters longer than its width. If the area of the garden is 135 square meters, find the dimensions of the garden by completing the square. Answer: 9 x 15 Solution: Let the width be x meters. Then the length is (x + 6) meters.
   Full step-by-step solution

   Step 1: Let the width be x meters. Then the length is (x + 6) meters. Step 2: The area equation is: x(x + 6) = 135 Step 3: Expand: x^2 + 6x = 135 Step 4: To complete the square, take half of the coefficient of x (which is 6), square it: (6/2)^2 = 3^2 = 9 Step 5: Add 9 to both sides: x^2 + 6x + 9 = 135 + 9 Step 6: Factor the left side: (x + 3)^2 = 144 Step 7: Take square root of both sides: x + 3 = ±12 Step 8: Since width cannot be negative: x + 3 = 12, so x = 9 Step 9: The length is x + 6 = 9 + 6 = 15 Step 10: The dimensions are 9 meters by 15 meters.

6. A square garden is drawn on a coordinate grid, with its bottom-left corner at (1, 1) and its top-right corner at (7, 7). A path of uniform width is added around the outside of the square, increasing the total area. The area of the path alone can be modeled by the quadratic expression 4x² + 32x, where x is the width of the path. Rewrite this expression in the form 4(x + h)² + k by first completing the square for the quadratic part (x² + 8x) inside the parentheses. What is the final expression? Answer: 4(x + 4)² - 64 Solution: Start with the expression 4x² + 32x. Factor out the common factor 4 from the terms involving x: 4(x² + 8x). Complete the square for the expression inside the parentheses: x² + 8x.
   Full step-by-step solution

   Step 1: Start with the expression 4x² + 32x. Factor out the common factor 4 from the terms involving x: 4(x² + 8x). Step 2: Complete the square for the expression inside the parentheses: x² + 8x. Take half of the coefficient of x (which is 8), so half is 4. Square it to get 16. Step 3: Add and subtract 16 inside the parentheses: x² + 8x + 16 - 16 = (x + 4)² - 16. Step 4: Multiply the completed square form by the factor 4: 4[(x + 4)² - 16] = 4(x + 4)² - 64. The final expression in the form 4(x + h)² + k is 4(x + 4)² - 64.

7. Noah is analyzing the graph of a parabola shown on a coordinate grid. The parabola opens upward and has its vertex at the point (7, -9). The graph passes through the point (9, -1). Write the equation of the parabola in the form y = (x - h)^2 + k. Answer: y = (x - 7)^2 - 9 Solution: Identify the vertex. The vertex is (7, -9), so h = 7 and k = -9. The equation starts as y = a(x - 7)^2 - 9.
   Full step-by-step solution

   Step 1: Identify the vertex. The vertex is (7, -9), so h = 7 and k = -9. The equation starts as y = a(x - 7)^2 - 9. Step 2: Use the point (9, -1) to find the value of a. Substitute x = 9 and y = -1 into the equation: -1 = a(9 - 7)^2 - 9 Step 3: Simplify inside the parentheses: -1 = a(2)^2 - 9 -1 = 4a - 9 Step 4: Solve for a. Add 9 to both sides: -1 + 9 = 4a 8 = 4a Step 5: Divide both sides by 4: a = 2 Step 6: Write the final equation: y = 2(x - 7)^2 - 9. The problem asks for the form y = (x - h)^2 + k, which means a = 1. Since a = 2, the parabola does not have a leading coefficient of 1. However, the vertex form is correctly written as y = 2(x - 7)^2 - 9. The answer is y = 2(x - 7)^2 - 9.

8. Isabella is designing a rectangular patio. The length of the patio is 7 meters more than the width. The area of the patio is 98 square meters. What is the width of the patio in meters? Answer: 7 Solution: Let the width be x meters. Since the length is 7 meters more than the width, the length is (x + 7) meters. The area is given as 98 square meters, so set up the equation: x(x + 7) = 98.
   Full step-by-step solution

   Step 1: Let the width be x meters. Since the length is 7 meters more than the width, the length is (x + 7) meters. Step 2: The area is given as 98 square meters, so set up the equation: x(x + 7) = 98. Step 3: Expand the equation: x^2 + 7x = 98. Step 4: Move the constant to the right side: x^2 + 7x = 98. Step 5: To complete the square, take half of the coefficient of x (which is 7), square it: (7/2)^2 = 49/4. Step 6: Add this value to both sides: x^2 + 7x + 49/4 = 98 + 49/4. Step 7: Write 98 as 392/4 to combine: 392/4 + 49/4 = 441/4. Step 8: Factor the left side as a perfect square: (x + 7/2)^2 = 441/4. Step 9: Take the square root of both sides: x + 7/2 = 21/2 or x + 7/2 = -21/2. Step 10: Solve for x: x = 21/2 - 7/2 = 14/2 = 7, or x = -21/2 - 7/2 = -28/2 = -14. Step 11: Since width cannot be negative, x = 7 meters. The answer is 7.