Complete the Square

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Answer Key & Explanations

Complete the Square · Grade 9 · Worksheet 2

1. 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.

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

   Let's solve this step-by-step. --- **Step 1: Define the 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 equation for area** Area of rectangle = length × width Given: \( l \times w = 96 \) Substitute \( l = w + 4 \): \[ (w + 4) \times w = 96 \] --- **Step 3: Expand and rearrange** \[ w^2 + 4w = 96 \] Subtract 96 from both sides: \[ w^2 + 4w - 96 = 0 \] --- **Step 4: Solve the quadratic equation** We can factor: Look for two numbers that multiply to -96 and add to 4. Those numbers are 12 and -8. So: \[ w^2 + 4w - 96 = (w + 12)(w - 8) = 0 \] --- **Step 5: Find possible values of \( w \)** \[ w + 12 = 0 \quad \text{or} \quad w - 8 = 0 \] \[ w = -12 \quad \text{or} \quad w = 8 \] --- **Step 6: Interpret the solutions** Width cannot be negative, so \( w = -12 \) is not valid. Thus, \( w = 8 \) meters. --- **Step 7: Check** Width = 8 m Length = 8 + 4 = 12 m Area = 12 × 8 = 96 m² ✔ --- **Final answer:** 8

4. Kaia is designing a rectangular skateboard ramp. The length of the ramp's base is 12 meters more than its width. The total area of the ramp's rectangular base is 220 square meters. Write the expression for the area in the form (x + h)^2 + k, where x represents the width of the ramp in meters. What is the value of k? Answer: k = -36 Solution: Let the width be x meters. Since the length is 12 meters more, the length is (x + 12) meters. Area = length * width = x(x + 12) = x^2 + 12x.
   Full step-by-step solution

   Step 1: Let the width be x meters. Since the length is 12 meters more, the length is (x + 12) meters. Step 2: Area = length * width = x(x + 12) = x^2 + 12x. Step 3: The area is given as 220 square meters, so we have x^2 + 12x = 220. Step 4: To complete the square on the left side, take half of 12, which is 6, and square it to get 36. Step 5: Add and subtract 36 on the left side: x^2 + 12x + 36 - 36 = 220. Step 6: This becomes (x + 6)^2 - 36 = 220. Step 7: The expression in the form (x + h)^2 + k is (x + 6)^2 - 36, where h = 6 and k = -36.

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (0,5). 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: 132.665 Solution: Identify the hypotenuse of the right triangle using the given vertices (0,0), (12,0), and (0,5). The hypotenuse is the side between (12,0) and (0,5).
   Full step-by-step solution

   Step 1: Identify the hypotenuse of the right triangle using the given vertices (0,0), (12,0), and (0,5). The hypotenuse is the side between (12,0) and (0,5). Step 2: Calculate the length of the hypotenuse using the distance formula: sqrt((12-0)^2 + (0-5)^2) = sqrt(144 + 25) = sqrt(169) = 13 units. Step 3: For a right triangle inscribed in a circle, the hypotenuse is the diameter of the circle. Therefore, the diameter d = 13 units. Step 4: Calculate the radius: r = d/2 = 13/2 = 6.5 units. Step 5: Calculate the area of the circle using A = πr^2 = 3.14 × (6.5)^2 = 3.14 × 42.25 = 132.665. The area of the circle is 132.665 square units.

6. Aroha is designing a rectangular vegetable patch. The length of the patch is 5 meters more than its width. The area of the patch is 84 square meters. By completing the square, express the relationship between the width and the area in the form (w + a)² = b, where w is the width in meters. Answer: (w + 2.5)² = 90.25 Solution: Let the width be w meters. Since the length is 5 meters more than the width, length = w + 5. Area = length × width = w(w + 5) = 84.
   Full step-by-step solution

   Step 1: Let the width be w meters. Since the length is 5 meters more than the width, length = w + 5. Step 2: Area = length × width = w(w + 5) = 84. Step 3: Expand: w² + 5w = 84. Step 4: To complete the square, take half of the coefficient of w (which is 5): 5/2 = 2.5. Square it: (2.5)² = 6.25. Step 5: Add 6.25 to both sides: w² + 5w + 6.25 = 84 + 6.25 = 90.25. Step 6: The left side is a perfect square: (w + 2.5)² = 90.25. The answer is (w + 2.5)² = 90.25.

7. 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.

8. 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, determine 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 from (x,0) to (0,6), length = 10 In a right triangle, the sum of squares of the two legs equals the square of the…
   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 from (x,0) to (0,6), length = 10 --- **Step 2: Apply the Pythagorean theorem** In a right triangle, the sum of squares of the two legs equals the square of the hypotenuse. Here, the legs are AB and AC. So: (AB)^2 + (AC)^2 = (BC)^2 Substitute the lengths: AB = x (since x > 0 for a triangle in the first quadrant) AC = 6 BC = 10 Equation: x^2 + 6^2 = 10^2 --- **Step 3: Simplify the equation** 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) = 8 (We take the positive root because the point (x,0) is to the right of the origin in the coordinate plane for the triangle to be drawn in the first quadrant.) --- **Final answer:** x = 8