Quadratic by Factoring

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Quadratic by Factoring · Grade 9 · Worksheet 2

1. A rectangular parking lot has an area of 180 square meters. The length of the parking lot is 3 meters less than twice its width. What are the dimensions of the parking lot in meters? Answer: 12, 15 Solution: Let w represent the width of the parking lot in meters. The length is 3 meters less than twice the width, so length = 2w - 3. The area is length × width = (2w - 3)(w) = 180.
   Full step-by-step solution

   Step 1: Let w represent the width of the parking lot in meters. Step 2: The length is 3 meters less than twice the width, so length = 2w - 3. Step 3: The area is length × width = (2w - 3)(w) = 180. Step 4: Expand the equation: 2w² - 3w = 180. Step 5: Subtract 180 from both sides: 2w² - 3w - 180 = 0. Step 6: Factor the quadratic equation: (2w + ?)(w - ?) = 0. Looking for factors of -360 that add to -3: -15 and 24 work. Step 7: Rewrite as (2w - 15)(w + 12) = 0. Step 8: Set each factor equal to zero: 2w - 15 = 0 or w + 12 = 0. Step 9: Solve for w: w = 15/2 = 7.5 or w = -12. Step 10: Since width cannot be negative, w = 7.5 meters. Step 11: Calculate length: length = 2(7.5) - 3 = 15 - 3 = 12 meters. The dimensions are 7.5 meters by 12 meters.

2. A rectangular garden has an area of 54 square meters. The length of the garden is 3 meters more than twice its width. Liam needs to build a fence around the entire garden. What are the dimensions of the garden in meters? Answer: width = 4.5 m, length = 12 m Solution: width = w meters length = l meters - Area = 54 m² - Length is 3 more than twice the width.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define variables** Let width = w meters length = l meters We are told: - Area = 54 m² - Length is 3 more than twice the width. So: l = 2w + 3 --- **Step 2: Write the area equation** Area = length × width 54 = l × w Substitute l from Step 1: 54 = (2w + 3) × w --- **Step 3: Expand and rearrange** 54 = 2w² + 3w Bring all terms to one side: 2w² + 3w - 54 = 0 --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: w = [ -b ± sqrt(b² - 4ac) ] / (2a) Here a = 2, b = 3, c = -54. Discriminant: b² - 4ac = (3)² - 4(2)(-54) = 9 + 432 = 441 sqrt(441) = 21 So: w = [ -3 ± 21 ] / (4) --- **Step 5: Two possible solutions** First: w = ( -3 + 21 ) / 4 = 18 / 4 = 4.5 Second: w = ( -3 - 21 ) / 4 = -24 / 4 = -6 Width cannot be negative, so w = 4.5 m. --- **Step 6: Find length** l = 2w + 3 l = 2(4.5) + 3 l = 9 + 3 l = 12 m --- **Step 7: Check area** Area = 12 × 4.5 = 54 m² ✓ Length = 12, which is 3 more than twice 4.5 (since 2×4.5 = 9, plus 3 = 12) ✓ --- **Final answer:** width = 4.5 m, length = 12 m

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,6). The area of the triangle is 12 square units. Write the quadratic equation in standard form that represents this situation, then solve for x by factoring. Answer: x=4 Solution: The vertices are (0,0), (x,0), and (0,6). This means the base is along the x-axis from (0,0) to (x,0), so base length = x. The height is along the y-axis from (0,0) to (0,6), so height = 6.
   Full step-by-step solution

   Step 1: Understand the triangle's vertices and area formula. The vertices are (0,0), (x,0), and (0,6). This means the base is along the x-axis from (0,0) to (x,0), so base length = x. The height is along the y-axis from (0,0) to (0,6), so height = 6. Step 2: Write the area formula for a right triangle. Area = (1/2) * base * height. Substitute the known values: 12 = (1/2) * x * 6. Step 3: Simplify the equation. 12 = (1/2) * 6 * x 12 = 3 * x Divide both sides by 3: x = 4. Step 4: Check if a quadratic equation is needed. The problem says: "Write the quadratic equation in standard form that represents this situation, then solve for x by factoring." But from our steps, we got x = 4 directly — no quadratic yet. This means we must have made an assumption: we assumed x > 0 so base = x. But if x is negative, base length would be |x|, but vertices (x,0) would be left of the origin, but the triangle's area formula still uses positive base length. Let’s re-derive carefully: Step 5: General area formula for a right triangle with legs along the axes. Vertices: (0,0), (a,0), (0,b) → area = (1/2) * |a| * |b|. Here b = 6, area = 12. So 12 = (1/2) * |x| * 6 12 = 3 * |x| |x| = 4 So x = 4 or x = -4. Step 6: Write as a quadratic equation. From |x| = 4, square both sides: x^2 = 16 So x^2 - 16 = 0 is the quadratic in standard form. Step 7: Solve by factoring. x^2 - 16 = (x - 4)(x + 4) = 0 So x = 4 or x = -4. Step 8: Interpret the solutions. Both x = 4 and x = -4 give base length 4 and height 6, area = 12. But the problem likely expects the positive solution x = 4 as the final answer. Final answer: x = 4.

4. Charlotte is building a rectangular garden in her backyard. The area of the garden is 120 square feet. The length of the garden is 7 feet more than its width. What are the dimensions of the garden in feet? Answer: 8 feet by 15 feet Solution: Let w represent the width in feet. Then the length is w + 7 feet. The area is length times width, so: w(w + 7) = 120.
   Full step-by-step solution

   Let w represent the width in feet. Then the length is w + 7 feet. The area is length times width, so: w(w + 7) = 120. Expand: w^2 + 7w = 120. Subtract 120 from both sides: w^2 + 7w - 120 = 0. Factor the quadratic: (w + 15)(w - 8) = 0. Apply the zero product property: w + 15 = 0 or w - 8 = 0, so w = -15 or w = 8. Since width cannot be negative, w = 8 feet. Then length = w + 7 = 8 + 7 = 15 feet. The dimensions are 8 feet by 15 feet.

5. x² - 8x - 48 = 0 Answer: x = 12, x = -4 Solution: Factor the quadratic x² - 8x - 48 = 0. Find two numbers that multiply to -48 and add to -8. The numbers are -12 and 4 because (-12) × 4 = -48 and (-12) + 4 = -8.
   Full step-by-step solution

   Step 1: Factor the quadratic x² - 8x - 48 = 0. Step 2: Find two numbers that multiply to -48 and add to -8. The numbers are -12 and 4 because (-12) × 4 = -48 and (-12) + 4 = -8. Step 3: Write the factored form: (x - 12)(x + 4) = 0. Step 4: Apply the zero product property: x - 12 = 0 or x + 4 = 0. Step 5: Solve each equation: x = 12 or x = -4. The solutions are x = 12 and x = -4.

6. Matiu is designing a rectangular fish pond on a coordinate grid, with one corner at the origin (0,0) and the adjacent sides lying along the positive x-axis and positive y-axis. The length of the pond (along the x-axis) is 14 meters more than the width (along the y-axis). The area of the rectangular pond is 207 square meters. Write a quadratic equation in standard form that represents this situation, then solve for the width by factoring. Answer: 9 Solution: Let w represent the width of the pond in meters. Since the length is 14 meters more than the width, the length is w + 14. The area of a rectangle is length times width.
   Full step-by-step solution

   Step 1: Let w represent the width of the pond in meters. Since the length is 14 meters more than the width, the length is w + 14. Step 2: The area of a rectangle is length times width. So area = (w + 14) * w = w^2 + 14w. Step 3: Set this equal to the given area of 207: w^2 + 14w = 207. Step 4: Subtract 207 from both sides to get standard form: w^2 + 14w - 207 = 0. Step 5: Factor the quadratic. Find two numbers that multiply to -207 and add to 14. These numbers are 23 and -9 because 23 * (-9) = -207 and 23 + (-9) = 14. Step 6: Write the factored form: (w + 23)(w - 9) = 0. Step 7: Apply the zero product property: w + 23 = 0 or w - 9 = 0. Step 8: Solve each equation: w = -23 or w = 9. Step 9: Since width cannot be negative, discard w = -23. The width is 9 meters. The answer is 9.

7. Emma is designing a rectangular flower bed for her school's garden. The area of the flower bed must be 143 square feet. The length of the flower bed is 2 feet more than its width. What are the dimensions of the flower bed in feet? Answer: 11 and 13 Solution: Let w represent the width of the flower bed in feet. Since the length is 2 feet more than the width, the length is w + 2 feet. The area of a rectangle is length times width, so the equation is w(w + 2) = 143.
   Full step-by-step solution

   Let w represent the width of the flower bed in feet. Since the length is 2 feet more than the width, the length is w + 2 feet. The area of a rectangle is length times width, so the equation is w(w + 2) = 143. Expand the left side: w^2 + 2w = 143. Subtract 143 from both sides to set the equation to zero: w^2 + 2w - 143 = 0. Now, factor the quadratic. We need two numbers that multiply to -143 and add to 2. The factors of 143 are 1 and 143, and 11 and 13. Since we need a product of -143, one factor must be negative. 13 and -11 multiply to -143 and add to 2. So the factored form is (w + 13)(w - 11) = 0. Apply the zero product property: w + 13 = 0 or w - 11 = 0. Solving gives w = -13 or w = 11. Since width cannot be negative, w = 11 feet. Then the length is w + 2 = 11 + 2 = 13 feet. The dimensions of the flower bed are 11 feet by 13 feet.

8. x² - 12x + 35 = 0 Answer: x = 5, 7 Solution: Identify the quadratic equation: x² - 12x + 35 = 0 Find two numbers that multiply to 35 and add to -12. Since the product is positive and the sum is negative, both numbers must be negative.
   Full step-by-step solution

   Step 1: Identify the quadratic equation: x² - 12x + 35 = 0 Step 2: Find two numbers that multiply to 35 and add to -12. Since the product is positive and the sum is negative, both numbers must be negative. The numbers are -5 and -7 because (-5) × (-7) = 35 and (-5) + (-7) = -12. Step 3: Write the factored form: (x - 5)(x - 7) = 0 Step 4: Apply the zero product property: x - 5 = 0 or x - 7 = 0 Step 5: Solve each equation: x = 5 or x = 7 The solutions are x = 5 and x = 7.