Quadratic Completing Square

lessonbunny.com

Answer Key & Explanations

Quadratic Completing Square · Grade 9 · Worksheet 2

1. x² + 6x - 7 = 0 Answer: x = 1, x = -7 Solution: x² + 6x - 7 = 0 Identify the coefficients. The equation is in the form: a x² + b x + c = 0 Here, a = 1, b = 6, c = -7. Factor the quadratic expression.
   Full step-by-step solution

   We are solving the quadratic equation: x² + 6x - 7 = 0 Step 1: Identify the coefficients. The equation is in the form: a x² + b x + c = 0 Here, a = 1, b = 6, c = -7. Step 2: Factor the quadratic expression. We look for two numbers that multiply to a * c = 1 * (-7) = -7 and add to b = 6. Possible pairs for -7: 1 and -7 → multiply to -7, add to -6 (no) -1 and 7 → multiply to -7, add to 6 (yes) So the numbers are -1 and 7. Step 3: Rewrite the middle term using these numbers. x² + 6x - 7 = x² - 1x + 7x - 7 Step 4: Factor by grouping. Group terms: (x² - 1x) + (7x - 7) Factor each group: x(x - 1) + 7(x - 1) Step 5: Factor out the common factor (x - 1). (x - 1)(x + 7) = 0 Step 6: Apply the zero-product property. If (x - 1)(x + 7) = 0, then: x - 1 = 0 or x + 7 = 0 Step 7: Solve each equation. x - 1 = 0 → x = 1 x + 7 = 0 → x = -7 Final answer: x = 1, x = -7

2. Aroha is designing a rectangular community garden. The length of the garden is 9 meters more than its width. The area of the garden is 112 square meters. Aroha wants to know the exact width and length of the garden so she can order fencing. Use completing the square to find the width and length of the garden. Answer: width = sqrt(193)/2 - 9/2 meters, length = sqrt(193)/2 + 9/2 meters Solution: Let x represent the width in meters. Then the length is x + 9 meters.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. Then the length is x + 9 meters. Step 2: Area = width × length = x(x + 9) = 112 Step 3: Expand: x² + 9x = 112 Step 4: To complete the square, take half of 9, which is 9/2, and square it: (9/2)² = 81/4 Step 5: Add 81/4 to both sides: x² + 9x + 81/4 = 112 + 81/4 Step 6: Convert 112 to fourths: 112 = 448/4, so 448/4 + 81/4 = 529/4 Step 7: The left side is a perfect square: (x + 9/2)² = 529/4 Step 8: Take the square root of both sides: x + 9/2 = ± sqrt(529/4) = ± sqrt(529)/2 = ± 23/2 Step 9: Solve for x: x = -9/2 ± 23/2 Step 10: Two possible solutions: x = (-9 + 23)/2 = 14/2 = 7, or x = (-9 - 23)/2 = -32/2 = -16 Step 11: Since width cannot be negative, x = 7 meters. Step 12: Length = x + 9 = 7 + 9 = 16 meters. Step 13: The width is 7 meters and the length is 16 meters. Answer: width = 7 meters, length = 16 meters

3. Emma is designing a rectangular flower bed for her school garden. The area of the flower bed can be modeled by the quadratic expression x² + 12x, where x represents the width in meters. To determine the dimensions that would make the flower bed a perfect square, Emma needs to rewrite the expression in vertex form by completing the square. What constant term must she add to x² + 12x to create a perfect square trinomial? Answer: 36 Solution: Start with the expression: x² + 12x To complete the square, take half of the coefficient of x: 12 ÷ 2 = 6 Square this result: 6² = 36 The perfect square trinomial is (x + 6)² = x² + 12x + 36 Therefore, the constant term needed is 36 The answer is 36.
   Full step-by-step solution

   Step 1: Start with the expression: x² + 12x Step 2: To complete the square, take half of the coefficient of x: 12 ÷ 2 = 6 Step 3: Square this result: 6² = 36 Step 4: The perfect square trinomial is (x + 6)² = x² + 12x + 36 Step 5: Therefore, the constant term needed is 36 The answer is 36.

4. Emma is designing a rectangular community garden. The length of the garden is 5 meters more than its width. The area of the garden is 150 square meters. To determine the exact dimensions, Emma needs to solve the quadratic equation by completing the square. What are the width and length of the garden? Answer: Width = 10 meters, Length = 15 meters Solution: Let x represent the width in meters. Then the length is x + 5 meters. The area is x(x + 5) = 150.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. Then the length is x + 5 meters. The area is x(x + 5) = 150. Step 2: Expand: x^2 + 5x = 150. Step 3: Move the constant to the left: x^2 + 5x - 150 = 0. Step 4: To complete the square, take half of the coefficient of x (which is 5), square it: (5/2)^2 = 25/4 = 6.25. Step 5: Add and subtract 25/4 to the left side: x^2 + 5x + 25/4 - 25/4 - 150 = 0. Step 6: Rewrite as a perfect square: (x + 5/2)^2 - 25/4 - 150 = 0. Step 7: Combine constants: -25/4 - 150 = -25/4 - 600/4 = -625/4. Step 8: So (x + 5/2)^2 - 625/4 = 0, then (x + 5/2)^2 = 625/4. Step 9: Take square root of both sides: x + 5/2 = ± sqrt(625/4) = ± 25/2. Step 10: Solve for x: x = -5/2 ± 25/2. Step 11: The two solutions are x = (-5/2 + 25/2) = 20/2 = 10, and x = (-5/2 - 25/2) = -30/2 = -15. Since width cannot be negative, x = 10 meters. Step 12: Length = x + 5 = 10 + 5 = 15 meters. The answer is: Width = 10 meters, Length = 15 meters.

5. Emma is designing a rectangular flower bed for her school's garden. The area of the flower bed can be modeled by the quadratic expression x² + 12x, where x represents the width in meters. To determine the dimensions that would make the flower bed a perfect square, she needs to rewrite the expression in vertex form by completing the square. What constant term must Emma add to x² + 12x to create a perfect square trinomial? Answer: 36 Solution: Start with the expression x² + 12x To complete the square, take half of the coefficient of x: 12 ÷ 2 = 6 Square this result: 6² = 36 The perfect square trinomial is (x + 6)² = x² + 12x + 36 Therefore, the constant term needed is 36 The answer is 36.
   Full step-by-step solution

   Step 1: Start with the expression x² + 12x Step 2: To complete the square, take half of the coefficient of x: 12 ÷ 2 = 6 Step 3: Square this result: 6² = 36 Step 4: The perfect square trinomial is (x + 6)² = x² + 12x + 36 Step 5: Therefore, the constant term needed is 36 The answer is 36.

6. A rectangular garden has a length that is 4 meters more than its width. If the area of the garden is 96 square meters, find the dimensions of the garden by setting up a quadratic equation and solving it by completing the square. Answer: width = 8 meters, length = 12 meters 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 go 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 area equation** Area = length × width \( 96 = (w + 4) \times w \) So: \( 96 = w^2 + 4w \) --- **Step 3: Rearrange into standard quadratic form** \( w^2 + 4w - 96 = 0 \) --- **Step 4: Solve by completing 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: \( 4/2 = 2 \), square it: \( 2^2 = 4 \). 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: \( w + 2 = -10 \) → \( w = -12 \) (discard, width can't be negative) --- **Step 8: Find length** \( l = w + 4 = 8 + 4 = 12 \) --- **Final answer:** Width = 8 meters, Length = 12 meters

7. Olivia is designing a rectangular koi pond for her backyard. The pond will have an area of 135 square meters. She wants the length to be 6 meters more than the width. To order the correct liner, she needs the exact dimensions of the pond. Set up a quadratic equation for this situation and solve it by completing the square to find the width and length of the pond. Answer: width = -3 + 3√15 meters, length = 3 + 3√15 meters Solution: Let x represent the width in meters. Then the length is x + 6 meters.
   Full step-by-step solution

   Step 1: Let x represent the width in meters. Then the length is x + 6 meters. Step 2: Area = width × length = x(x + 6) = 135 Step 3: Expand: x² + 6x = 135 Step 4: Move the constant to the right side: x² + 6x = 135 Step 5: Complete the square: take half of 6, which is 3, then square it to get 9. Add 9 to both sides: x² + 6x + 9 = 135 + 9 Step 6: The left side is now a perfect square trinomial: (x + 3)² = 144 Step 7: Take the square root of both sides: x + 3 = ±√144 Step 8: Simplify: x + 3 = ±12 Step 9: Solve for x: x + 3 = 12 → x = 9 x + 3 = -12 → x = -15 (reject, width cannot be negative) Step 10: So width = 9 meters, length = 9 + 6 = 15 meters. Final answer: width = 9 meters, length = 15 meters.