Complete the Square

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Answer Key & Explanations

Complete the Square · Grade 9 · Worksheet 2

1. 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.

2. 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.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,6). The hypotenuse has length 10 units. Using the Pythagorean theorem, find the value of x. Answer: 8 Solution: A = (0, 0) B = (x, 0) C = (0, 6) - AB is along the x-axis from (0,0) to (x,0), length = x - AC is along the y-axis from (0,0) to (0,6), length = 6 - BC is the hypotenuse, length = 10 (leg1)^2 + (leg2)^2 = (hypotenuse)^2 leg1 = AB = x leg2 = AC = 6 hypotenuse = BC = 10 x^2 + 6^2 = 10^2 x^2 + 36 =…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle's vertices** The vertices are: A = (0, 0) B = (x, 0) C = (0, 6) So: - AB is along the x-axis from (0,0) to (x,0), length = x - AC is along the y-axis from (0,0) to (0,6), length = 6 - BC is the hypotenuse, length = 10 --- **Step 2: Apply the Pythagorean theorem** The Pythagorean theorem says: (leg1)^2 + (leg2)^2 = (hypotenuse)^2 Here: leg1 = AB = x leg2 = AC = 6 hypotenuse = BC = 10 So: x^2 + 6^2 = 10^2 --- **Step 3: Substitute numbers** x^2 + 36 = 100 --- **Step 4: Solve for x^2** x^2 = 100 - 36 x^2 = 64 --- **Step 5: Solve for x** x = sqrt(64) or x = -sqrt(64) x = 8 or x = -8 --- **Step 6: Interpret the solution** Since x is a length in this geometry problem, we take the positive value: x = 8 --- **Final answer:** 8

4. 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.

5. x² - 4x - 5 = 0 Answer: x = 5, -1 Solution: Start with x² - 4x - 5 = 0 Move constant term to right side: x² - 4x = 5 Find the number to complete the square: take half of -4, which is -2, then square it to get 4 Add 4 to both sides: x² - 4x + 4 = 5 + 4 Write left side as perfect square: (x - 2)² = 9 Take square root of both sides: x - 2 =…
   Full step-by-step solution

   Step 1: Start with x² - 4x - 5 = 0 Step 2: Move constant term to right side: x² - 4x = 5 Step 3: Find the number to complete the square: take half of -4, which is -2, then square it to get 4 Step 4: Add 4 to both sides: x² - 4x + 4 = 5 + 4 Step 5: Write left side as perfect square: (x - 2)² = 9 Step 6: Take square root of both sides: x - 2 = ±3 Step 7: Solve for x: x = 2 + 3 = 5 or x = 2 - 3 = -1 The solutions are x = 5 and x = -1.

6. x² - 10x + 21 = (x - ?)² - 4 Answer: 5 Solution: Start with x² - 10x + 21 To complete the square, take half of the coefficient of x: (-10)/2 = -5 Square this result: (-5)² = 25 Add and subtract 25: x² - 10x + 25 - 25 + 21 Group the perfect square: (x² - 10x + 25) + (-25 + 21) Factor the perfect square: (x - 5)² - 4 Compare with the given form…
   Full step-by-step solution

   Step 1: Start with x² - 10x + 21 Step 2: To complete the square, take half of the coefficient of x: (-10)/2 = -5 Step 3: Square this result: (-5)² = 25 Step 4: Add and subtract 25: x² - 10x + 25 - 25 + 21 Step 5: Group the perfect square: (x² - 10x + 25) + (-25 + 21) Step 6: Factor the perfect square: (x - 5)² - 4 Step 7: Compare with the given form (x - ?)² - 4 Step 8: The missing value is 5 The answer is 5.

7. A square with side length x units is drawn on a coordinate plane. On the right side of the square, a rectangle of width 6 units is attached, forming a larger rectangle. The total area of the combined shape is represented by the expression x² + 6x. If the expression is rewritten in the form (x + h)² + k, what is the value of k? Answer: -9 Solution: Start with the expression x² + 6x. To complete the square, take half of the coefficient of x, which is 6/2 = 3. Square this value: 3² = 9.
   Full step-by-step solution

   Step 1: Start with the expression x² + 6x. Step 2: To complete the square, take half of the coefficient of x, which is 6/2 = 3. Step 3: Square this value: 3² = 9. Step 4: Add and subtract 9 to the expression: x² + 6x + 9 - 9. Step 5: Group the perfect square trinomial: (x² + 6x + 9) - 9. Step 6: Factor the trinomial: (x + 3)² - 9. Step 7: This is in the form (x + h)² + k, where h = 3 and k = -9. The value of k is -9.

8. Aroha is designing a rectangular community garden. The length of the garden is 14 meters more than its width. The area of the garden is 315 square meters. Rewrite the quadratic expression representing the area in the form (w + h)^2 + k, where w is the width of the garden. Answer: (w + 7)^2 - 364 Solution: Let w represent the width of the garden 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 represent the width of the garden in meters. Since the length is 14 meters more than the width, the length is (w + 14) meters. Step 2: The area of a rectangle is length times width, so the area expression is w(w + 14) = w^2 + 14w. Step 3: We need to rewrite w^2 + 14w in the form (w + h)^2 + k. Step 4: Take half of the coefficient of w: 14 / 2 = 7. Step 5: Square this value: 7^2 = 49. Step 6: Add and subtract 49: w^2 + 14w + 49 - 49. Step 7: Factor the perfect square trinomial: (w + 7)^2 - 49. Step 8: The area of the garden is given as 315, but the question asks to rewrite the expression for area, not to solve. The expression w^2 + 14w completed to square form is (w + 7)^2 - 49. Step 9: However, the problem states the area is 315, so we are working with the equation w^2 + 14w = 315. To rewrite the quadratic expression w^2 + 14w in completed square form, we use the same process: w^2 + 14w = (w + 7)^2 - 49. Step 10: Therefore, the expression in the form (w + h)^2 + k is (w + 7)^2 - 49. The answer is (w + 7)^2 - 49.