Quadratic by Factoring

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Answer Key & Explanations

Quadratic by Factoring · Grade 9 · Worksheet 3

1. Liam is designing a rectangular garden with a path of uniform width around it. The garden itself measures 12 meters by 8 meters. The total area of the garden plus the path is 140 square meters. What is the width of the path in meters? Answer: 1 Solution: The garden is a rectangle 12 m by 8 m. A path of uniform width \( x \) meters surrounds it. Total area (garden + path) = 140 m².
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The garden is a rectangle 12 m by 8 m. A path of uniform width \( x \) meters surrounds it. Total area (garden + path) = 140 m². --- **Step 2: Find dimensions of the outer rectangle (including path)** The path is on all sides, so it adds \( x \) meters to both the length and the width. Length of outer rectangle = \( 12 + 2x \) Width of outer rectangle = \( 8 + 2x \) --- **Step 3: Set up the area equation** Area of outer rectangle = \( (12 + 2x)(8 + 2x) \) We know this equals 140. So: \( (12 + 2x)(8 + 2x) = 140 \) --- **Step 4: Expand the equation** First, multiply: \( 12 \cdot 8 = 96 \) \( 12 \cdot 2x = 24x \) \( 2x \cdot 8 = 16x \) \( 2x \cdot 2x = 4x^2 \) Add them: \( 96 + 24x + 16x + 4x^2 = 140 \) \( 96 + 40x + 4x^2 = 140 \) --- **Step 5: Simplify** Subtract 140 from both sides: \( 96 + 40x + 4x^2 - 140 = 0 \) \( 4x^2 + 40x - 44 = 0 \) --- **Step 6: Divide through by 4** \( x^2 + 10x - 11 = 0 \) --- **Step 7: Solve the quadratic equation** Factor: \( x^2 + 10x - 11 = 0 \) We need two numbers that multiply to -11 and add to 10: They are 11 and -1. So: \( (x + 11)(x - 1) = 0 \) --- **Step 8: Find possible solutions** \( x + 11 = 0 \) → \( x = -11 \) (not possible, width can't be negative) \( x - 1 = 0 \) → \( x = 1 \) --- **Step 9: Check** If \( x = 1 \), outer dimensions: Length = \( 12 + 2(1) = 14 \) Width = \( 8 + 2(1) = 10 \) Area = \( 14 \times 10 = 140 \) m² ✔ --- **Final Answer:** The width of the path is **1 meter**.

2. Kaia is designing a rectangular community garden. The area of the garden is 180 square meters. The length of the garden is 3 meters more than its width. What are the dimensions of the garden in meters? Answer: 12 and 15 Solution: Let w represent the width in meters. Then the length is w + 3 meters. The area is length times width, so: w(w + 3) = 180.
   Full step-by-step solution

   Let w represent the width in meters. Then the length is w + 3 meters. The area is length times width, so: w(w + 3) = 180. Expand: w^2 + 3w = 180. Subtract 180 from both sides: w^2 + 3w - 180 = 0. Factor the quadratic: (w + 15)(w - 12) = 0. Apply the zero product property: w + 15 = 0 or w - 12 = 0, so w = -15 or w = 12. Since width cannot be negative, w = 12 meters. Then length = w + 3 = 12 + 3 = 15 meters. The dimensions are 12 meters by 15 meters.

3. A rectangular garden has an area of 54 square meters. The length of the garden is 3 meters more than twice its width. What are the dimensions of the garden? Answer: width = 4.5 meters, length = 12 meters Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \text{length} = 2w + 3 Area of rectangle = length × width (2w + 3) \times w = 54 2w^2 + 3w = 54 2w^2 + 3w - 54 = 0 For \( a w^2 + b w + c = 0 \), \( a = 2 \), \( b = 3 \), \( c = -54 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \[ \text{length} = 2w + 3 \] --- **Step 2: Write the area equation** Area of rectangle = length × width \[ (2w + 3) \times w = 54 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 54 \] \[ 2w^2 + 3w - 54 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: For \( a w^2 + b w + c = 0 \), \( a = 2 \), \( b = 3 \), \( c = -54 \). \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ w = \frac{-3 \pm \sqrt{3^2 - 4(2)(-54)}}{2 \times 2} \] \[ w = \frac{-3 \pm \sqrt{9 + 432}}{4} \] \[ w = \frac{-3 \pm \sqrt{441}}{4} \] \[ w = \frac{-3 \pm 21}{4} \] --- **Step 5: Two possible solutions** First: \( w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \) Second: \( w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \) Width cannot be negative, so \( w = 4.5 \) meters. --- **Step 6: Find length** \[ \text{length} = 2w + 3 = 2(4.5) + 3 = 9 + 3 = 12 \ \text{meters} \] --- **Step 7: Check** Area = \( 12 \times 4.5 = 54 \) square meters. Length \( 12 \) is indeed 3 more than twice the width \( 4.5 \) (since \( 2 \times 4.5 = 9 \), plus 3 is 12). --- **Final answer:** width = 4.5 meters, length = 12 meters

4. Olivia is designing a square garden with side length x meters. She plans to add a rectangular flower bed that extends x meters from one corner along the bottom edge and x+1 meters up the left edge, forming a right triangle flower bed with the corner of the garden. The area of the triangular flower bed is 15 square meters. Write a quadratic equation for x and solve it by factoring. Answer: 5 Solution: The triangular flower bed has legs of lengths x and x+1. The area of a right triangle is (1/2)*(base)*(height). Area = (1/2)*x*(x+1).
   Full step-by-step solution

   Step 1: The triangular flower bed has legs of lengths x and x+1. The area of a right triangle is (1/2)*(base)*(height). Area = (1/2)*x*(x+1). Step 2: Set this equal to 15: (1/2)*x*(x+1) = 15. Step 3: Multiply both sides by 2: x*(x+1) = 30. Step 4: Expand: x^2 + x = 30. Step 5: Subtract 30 from both sides to get standard form: x^2 + x - 30 = 0. Step 6: Factor the quadratic: Find two numbers that multiply to -30 and add to 1. These numbers are 6 and -5. So (x + 6)(x - 5) = 0. Step 7: Apply the zero product property: x + 6 = 0 or x - 5 = 0. Step 8: Solve: x = -6 or x = 5. Step 9: Since x is a side length, it must be positive. Discard x = -6. The answer is 5.

5. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden is 65 square meters. If you were to draw this garden on a coordinate plane with one corner at the origin (0,0) and the sides along the axes, what would be the dimensions of the rectangle? Answer: width = 5 m, length = 13 m Solution: Let the width of the rectangle be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the rectangle be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area = length × width \( l \times w = 65 \) Substitute \( l = 2w + 3 \): \( (2w + 3) \times w = 65 \). --- **Step 3: Expand and rearrange** \( 2w^2 + 3w = 65 \) Subtract 65 from both sides: \( 2w^2 + 3w - 65 = 0 \). --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here \( a = 2 \), \( b = 3 \), \( c = -65 \). Discriminant: \( b^2 - 4ac = 9 - 4(2)(-65) = 9 + 520 = 529 \). \( \sqrt{529} = 23 \). So: \( w = \frac{-3 \pm 23}{4} \). --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \). Case 2: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \). Width cannot be negative, so \( w = 5 \). --- **Step 6: Find length** \( l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \). --- **Step 7: Check** Area = \( 5 \times 13 = 65 \) m². Length 13 is indeed 3 more than twice the width (2×5 = 10, plus 3 is 13). --- **Final answer:** width = 5 m, length = 13 m.

6. Hana is designing a rectangular solar panel. The area of the solar panel is 288 square centimeters. The length of the panel is 12 centimeters more than its width. What are the dimensions of the solar panel in centimeters? Answer: 12 and 24 Solution: Let w represent the width in centimeters. Then the length is w + 12 centimeters. The area is length times width, so: w(w + 12) = 288.
   Full step-by-step solution

   Let w represent the width in centimeters. Then the length is w + 12 centimeters. The area is length times width, so: w(w + 12) = 288. Expand: w^2 + 12w = 288. Subtract 288 from both sides: w^2 + 12w - 288 = 0. Factor the quadratic: (w + 24)(w - 12) = 0. Apply the zero product property: w + 24 = 0 or w - 12 = 0. So w = -24 or w = 12. Since width cannot be negative, w = 12 cm. Then length = w + 12 = 12 + 12 = 24 cm. The dimensions are 12 cm by 24 cm.