Quadratic by Factoring

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Answer Key & Explanations

Quadratic by Factoring · Grade 9 · Worksheet 3

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,x+5). The area of the triangle is 42 square units. Write a quadratic equation in standard form that represents this situation and solve for x by factoring. Answer: 7 Solution: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (x,0), so base = x. The height is along the y-axis from (0,0) to (0,x+5), so height = x+5.
   Full step-by-step solution

   Step 1: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (x,0), so base = x. The height is along the y-axis from (0,0) to (0,x+5), so height = x+5. Step 2: Apply the area formula for a triangle: Area = (1/2) * base * height. Step 3: Substitute the known values: 42 = (1/2) * x * (x+5). Step 4: Multiply both sides by 2 to eliminate the fraction: 84 = x(x+5). Step 5: Expand the right side: 84 = x^2 + 5x. Step 6: Write the quadratic equation in standard form: x^2 + 5x - 84 = 0. Step 7: Factor the quadratic: (x + 12)(x - 7) = 0. Step 8: Solve for x: x = -12 or x = 7. Step 9: Since length cannot be negative, discard x = -12. The answer is 7.

2. A technology company is designing a new smartphone with a rectangular screen. The screen's area is 96 square centimeters, and its length is 4 centimeters greater than its width. The engineers need to calculate the exact dimensions of the screen for manufacturing. What are the width and length of the screen in centimeters? Answer: 8 and 12 Solution: Let the width be w centimeters. Then the length is w + 4 centimeters.
   Full step-by-step solution

   Step 1: Let the width be w centimeters. Then the length is w + 4 centimeters. Step 2: The area equation is: w(w + 4) = 96 Step 3: Expand the equation: w^2 + 4w = 96 Step 4: Subtract 96 from both sides: w^2 + 4w - 96 = 0 Step 5: Factor the quadratic: (w + 12)(w - 8) = 0 Step 6: Solve for w: w + 12 = 0 gives w = -12 (reject negative width) Step 7: w - 8 = 0 gives w = 8 Step 8: The length is w + 4 = 8 + 4 = 12 Step 9: The dimensions are width = 8 cm and length = 12 cm.

3. x² + 13x + 42 = 0 Answer: x = -6, -7 Solution: Factor the quadratic equation x² + 13x + 42 = 0 Find two numbers that multiply to 42 and add to 13 The numbers are 6 and 7 since 6 × 7 = 42 and 6 + 7 = 13 Write the factored form: (x + 6)(x + 7) = 0 Apply the zero product property: x + 6 = 0 or x + 7 = 0 Solve each equation: x = -6 or x = -7 The…
   Full step-by-step solution

   Step 1: Factor the quadratic equation x² + 13x + 42 = 0 Step 2: Find two numbers that multiply to 42 and add to 13 Step 3: The numbers are 6 and 7 since 6 × 7 = 42 and 6 + 7 = 13 Step 4: Write the factored form: (x + 6)(x + 7) = 0 Step 5: Apply the zero product property: x + 6 = 0 or x + 7 = 0 Step 6: Solve each equation: x = -6 or x = -7 Step 7: The solutions are x = -6 and x = -7

4. 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**.

5. A rectangular garden has an area of 54 square meters. The length of the garden is 3 meters more than twice its width. Find the dimensions of the garden in meters. Answer: width = 4.5, length = 12 Solution: Let the width of the garden 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 garden 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 of rectangle = length × width Given area = 54 m², so \( l \times w = 54 \) Substitute \( l = 2w + 3 \): \( (2w + 3) \times w = 54 \) --- **Step 3: Expand and rearrange** \( 2w^2 + 3w = 54 \) Subtract 54 from both sides: \( 2w^2 + 3w - 54 = 0 \) --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here \( a = 2 \), \( b = 3 \), \( c = -54 \). First, discriminant: \( D = b^2 - 4ac = 3^2 - 4(2)(-54) \) \( D = 9 + 432 = 441 \) --- **Step 5: Take square root of discriminant** \( \sqrt{D} = \sqrt{441} = 21 \) --- **Step 6: Apply quadratic formula** \( w = \frac{-3 \pm 21}{2 \times 2} \) \( w = \frac{-3 \pm 21}{4} \) Two possible solutions: \( w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \) \( w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \) (discard, width can't be negative) So \( w = 4.5 \) meters. --- **Step 7: Find length** \( l = 2w + 3 = 2(4.5) + 3 = 9 + 3 = 12 \) meters. --- **Step 8: Check** Area = \( 12 \times 4.5 = 54 \) m² ✔ Length = 12, which is 3 more than twice 4.5 (since twice 4.5 is 9, plus 3 is 12) ✔ --- **Final answer:** width = 4.5, length = 12

6. Charlotte is constructing a rectangular deck in her backyard. The area of the deck is 252 square feet. The length of the deck is 4 feet less than twice its width. What are the dimensions of the deck in feet? Answer: 14 feet by 18 feet Solution: Let w represent the width of the deck in feet. Then the length is 2w - 4 feet. The area is length times width, so w(2w - 4) = 252.
   Full step-by-step solution

   Let w represent the width of the deck in feet. Then the length is 2w - 4 feet. The area is length times width, so w(2w - 4) = 252. Expand to get 2w^2 - 4w = 252. Subtract 252 from both sides: 2w^2 - 4w - 252 = 0. Divide the entire equation by 2 to simplify: w^2 - 2w - 126 = 0. Factor the quadratic: (w - 14)(w + 9) = 0. Apply the zero product property: w - 14 = 0 or w + 9 = 0, so w = 14 or w = -9. Since width cannot be negative, w = 14 feet. Then length = 2(14) - 4 = 28 - 4 = 18 feet. The dimensions are 14 feet by 18 feet.