Quadratic Completing Square

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Quadratic Completing Square · Grade 9 · Worksheet 2

1. Emma is designing a square fountain for a park. She draws the fountain on a coordinate grid with one corner at the origin (0,0). The side length of the square is x meters. She then extends the fountain by adding a rectangular section along one side, increasing the total length by 9 meters, so the new shape is a rectangle with dimensions x by (x + 9). The area of this new rectangle is 112 square meters. Write a quadratic equation for this situation and solve for x by completing the square. Answer: 7 Solution: Write the area equation. The rectangle has width x and length x + 9. Area = x(x + 9) = 112.
   Full step-by-step solution

   Step 1: Write the area equation. The rectangle has width x and length x + 9. Area = x(x + 9) = 112. Step 2: Expand: x^2 + 9x = 112. Step 3: Move the constant to the left: x^2 + 9x - 112 = 0. Step 4: To complete the square, take half of the coefficient of x (which is 9), square it: (9/2)^2 = 81/4 = 20.25. Step 5: Add 81/4 to both sides of the equation x^2 + 9x = 112: x^2 + 9x + 81/4 = 112 + 81/4. Step 6: Write the left side as a perfect square: (x + 9/2)^2 = 112 + 81/4. Step 7: Convert 112 to a fraction with denominator 4: 112 = 448/4. So, 448/4 + 81/4 = 529/4. Step 8: (x + 9/2)^2 = 529/4. Step 9: Take the square root of both sides: x + 9/2 = ± sqrt(529/4) = ± 23/2. Step 10: Solve for x: x = -9/2 + 23/2 = 14/2 = 7, or x = -9/2 - 23/2 = -32/2 = -16. Step 11: Since x represents a length, it must be positive. Therefore, x = 7. The answer is 7.

2. x² + 10x + 18 = 0 Answer: -5 ± √7 Solution: Move the constant term to the right side: x² + 10x = -18 Find the number to complete the square: (10/2)² = 25 Add 25 to both sides: x² + 10x + 25 = -18 + 25 Factor the perfect square trinomial: (x + 5)² = 7 Take square root of both sides: x + 5 = ±√7 Solve for x: x = -5 ± √7 The answer is -5 ± √7.
   Full step-by-step solution

   Step 1: Move the constant term to the right side: x² + 10x = -18 Step 2: Find the number to complete the square: (10/2)² = 25 Step 3: Add 25 to both sides: x² + 10x + 25 = -18 + 25 Step 4: Factor the perfect square trinomial: (x + 5)² = 7 Step 5: Take square root of both sides: x + 5 = ±√7 Step 6: Solve for x: x = -5 ± √7 The answer is -5 ± √7.

3. 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

4. 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.

5. Hana is designing a square patio that will be surrounded by a uniform border of paving stones. The total area of the patio and the border together is 196 square units. If the side length of the square patio itself is x units and the border adds 4 units to each side, write a quadratic equation for this situation and solve for x by completing the square. Answer: 6 Solution: The patio is a square with side length x. The border adds 4 units to each side, so the side length of the large square (patio + border) is x + 8. Area of large square = (side)^2 = (x + 8)^2 = 196.
   Full step-by-step solution

   Step 1: The patio is a square with side length x. The border adds 4 units to each side, so the side length of the large square (patio + border) is x + 8. Step 2: Area of large square = (side)^2 = (x + 8)^2 = 196. Step 3: Expand: x^2 + 16x + 64 = 196. Step 4: Subtract 64 from both sides: x^2 + 16x = 132. Step 5: Complete the square: take half of 16, which is 8, then square it to get 64. Add 64 to both sides: x^2 + 16x + 64 = 132 + 64. Step 6: This gives x^2 + 16x + 64 = 196. Step 7: Factor the left side as a perfect square: (x + 8)^2 = 196. Step 8: Take the square root of both sides: x + 8 = ±14. Step 9: Solve for x: x = -8 + 14 = 6 or x = -8 - 14 = -22. Step 10: Since a side length cannot be negative, x = 6. The answer is 6.

6. Mason is designing a rectangular fountain for a city plaza. The fountain's length is 8 meters more than its width. The area of the fountain is 84 square meters. To ensure the fountain fits perfectly in the plaza, Mason needs to find the exact dimensions. Set up a quadratic equation and solve it by completing the square to determine the width and length of the fountain. Answer: width = 2√37 - 4 meters, length = 2√37 + 4 meters Solution: Let w represent the width in meters. The length is 8 meters more than the width, so length = w + 8. Area = width × length = w(w + 8) = 84.
   Full step-by-step solution

   Step 1: Let w represent the width in meters. Step 2: The length is 8 meters more than the width, so length = w + 8. Step 3: Area = width × length = w(w + 8) = 84. Step 4: Expand: w² + 8w = 84. Step 5: To complete the square, take half of 8, which is 4, and square it: 4² = 16. Step 6: Add 16 to both sides: w² + 8w + 16 = 84 + 16. Step 7: Simplify: w² + 8w + 16 = 100. Step 8: Rewrite the left side as a perfect square: (w + 4)² = 100. Step 9: Take the square root of both sides: w + 4 = ±√100. Step 10: √100 = 10, so w + 4 = 10 or w + 4 = -10. Step 11: Solve for w: w = 10 - 4 = 6 or w = -10 - 4 = -14. Step 12: Since width cannot be negative, w = 6 meters. Step 13: Length = w + 8 = 6 + 8 = 14 meters. The dimensions of the fountain are 6 meters wide and 14 meters long.

7. A rectangular garden has an area of 24 square meters. The length is 2 meters more than the width. Find the width of the garden by solving the quadratic equation x² + 2x - 24 = 0 using the completing the square method. Answer: 4 Solution: Write the equation. x² + 2x - 24 = 0 Move the constant term to the right side. Add 24 to both sides: x² + 2x = 24 Prepare to complete the square.
   Full step-by-step solution

   Let's solve the quadratic equation x² + 2x - 24 = 0 by completing the square. Step 1: Write the equation. x² + 2x - 24 = 0 Step 2: Move the constant term to the right side. Add 24 to both sides: x² + 2x = 24 Step 3: Prepare to complete the square. We have x² + 2x. To complete the square, take half of the coefficient of x, then square it. Coefficient of x is 2. Half of 2 is 1. Square of 1 is 1. Step 4: Add this square to both sides of the equation. x² + 2x + 1 = 24 + 1 x² + 2x + 1 = 25 Step 5: Factor the left side (it's now a perfect square trinomial). (x + 1)² = 25 Step 6: Take the square root of both sides. Remember: when taking square roots, we must consider both positive and negative roots. x + 1 = 5 or x + 1 = -5 Step 7: Solve each equation for x. First equation: x + 1 = 5 → x = 5 - 1 → x = 4 Second equation: x + 1 = -5 → x = -5 - 1 → x = -6 Step 8: Interpret the solutions. Since x represents the width of a garden, it cannot be negative. Therefore, we discard x = -6. The width of the garden is x = 4 meters. Final answer: 4