Quadratic by Factoring

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Quadratic by Factoring · Grade 9 · Worksheet 1

1. A technology company is designing a new smartphone with a rectangular screen. The screen's area is 216 square centimeters, and its length is 6 centimeters more than its width. The engineers need to calculate the exact dimensions to determine the phone's overall size. What are the dimensions of the screen in centimeters? Answer: 12 and 18 Solution: In problems involving rectangular areas with given relationships between dimensions, we can use variables to represent the unknown quantities. The area formula for a rectangle (length × width = area) combined with the given relationship allows us to set up a quadratic equation.
   Full step-by-step solution

   In problems involving rectangular areas with given relationships between dimensions, we can use variables to represent the unknown quantities. The area formula for a rectangle (length × width = area) combined with the given relationship allows us to set up a quadratic equation. Factoring this equation helps us find the possible dimensions that satisfy both conditions.

2. Mason is designing a rectangular fountain on a coordinate grid. One corner of the fountain is at the origin (0,0), and the fountain extends along the positive x-axis and positive y-axis. The length along the x-axis is 8 meters more than the width along the y-axis. The area of the fountain is 84 square meters. Write a quadratic equation in standard form that represents this situation, then solve for the width by factoring. Answer: 6 Solution: Let w represent the width of the fountain along the y-axis (in meters). Since the length along the x-axis is 8 meters more than the width, the length is w + 8. The area of a rectangle is length times width.
   Full step-by-step solution

   Step 1: Let w represent the width of the fountain along the y-axis (in meters). Since the length along the x-axis is 8 meters more than the width, the length is w + 8. Step 2: The area of a rectangle is length times width. So the area is (w + 8) * w = w^2 + 8w. Step 3: Set this equal to the given area of 84: w^2 + 8w = 84. Step 4: Subtract 84 from both sides to get standard form: w^2 + 8w - 84 = 0. Step 5: Factor the quadratic. We need two numbers that multiply to -84 and add to 8. The factors of 84 are 1, 84; 2, 42; 3, 28; 4, 21; 6, 14; 7, 12. The pair 14 and -6 multiply to -84 and add to 8 (because 14 + (-6) = 8). Step 6: Write the factored form: (w + 14)(w - 6) = 0. Step 7: Apply the zero product property: w + 14 = 0 or w - 6 = 0. Step 8: Solve each equation: w = -14 or w = 6. Step 9: Since width cannot be negative, discard w = -14. The width is 6 meters. The answer is 6.

3. Mere is designing a rectangular patio that will be drawn on a coordinate grid. One corner is at the origin (0,0), and the adjacent sides lie along the positive x-axis and positive y-axis. The length along the x-axis is x meters, and the length along the y-axis is (x + 2) meters. The area of the rectangle is 48 square meters. Write a quadratic equation in standard form that represents this situation, then solve for x by factoring to find the dimensions of the patio. Answer: 6 Solution: The rectangle has vertices at (0,0), (x,0), (x,x+2), and (0,x+2). The base along the x-axis has length x, and the height along the y-axis has length (x+2). The area of a rectangle is base times height.
   Full step-by-step solution

   Step 1: The rectangle has vertices at (0,0), (x,0), (x,x+2), and (0,x+2). The base along the x-axis has length x, and the height along the y-axis has length (x+2). Step 2: The area of a rectangle is base times height. So area = x * (x+2). Step 3: Set this equal to the given area of 48: x(x+2) = 48. Step 4: Expand the left side: x^2 + 2x = 48. Step 5: Subtract 48 from both sides to get standard form: x^2 + 2x - 48 = 0. Step 6: Factor the quadratic. Find two numbers that multiply to -48 and add to 2. These numbers are 8 and -6. Step 7: Write the factored form: (x + 8)(x - 6) = 0. Step 8: Apply the zero product property: x + 8 = 0 or x - 6 = 0. Step 9: Solve for x: x = -8 or x = 6. Step 10: Since length cannot be negative, discard x = -8. So x = 6. The dimensions of the patio are 6 meters by 8 meters. The answer is 6.

4. Sophia is designing a rectangular banner for a school event. The area of the banner is 280 square inches. The length of the banner is 6 inches more than twice its width. What are the dimensions of the banner in inches? Answer: 10 and 26 Solution: Let w represent the width in inches. Then the length is 2w + 6. The area is length times width, so w(2w + 6) = 280.
   Full step-by-step solution

   Let w represent the width in inches. Then the length is 2w + 6. The area is length times width, so w(2w + 6) = 280. Expand: 2w^2 + 6w = 280. Subtract 280 from both sides: 2w^2 + 6w - 280 = 0. Divide the entire equation by 2 to simplify: w^2 + 3w - 140 = 0. Factor the quadratic: (w + 14)(w - 10) = 0. Apply the zero product property: w + 14 = 0 or w - 10 = 0, so w = -14 or w = 10. Since width cannot be negative, w = 10 inches. Then length = 2(10) + 6 = 20 + 6 = 26 inches. The dimensions are 10 inches by 26 inches.

5. Mere is designing a rectangular patio for her backyard. The area of the patio is 140 square feet. The length of the patio is 4 feet less than three times its width. What are the dimensions of the patio in feet? Answer: 7 and 17 Solution: Let w represent the width in feet. Then the length is 3w - 4 feet. The area is length times width, so: w(3w - 4) = 140.
   Full step-by-step solution

   Let w represent the width in feet. Then the length is 3w - 4 feet. The area is length times width, so: w(3w - 4) = 140. Expand: 3w^2 - 4w = 140. Subtract 140 from both sides: 3w^2 - 4w - 140 = 0. Factor the quadratic: (3w + 14)(w - 10) = 0. Apply the zero product property: 3w + 14 = 0 or w - 10 = 0. Solve: 3w = -14 so w = -14/3, or w = 10. Since width cannot be negative, w = 10 feet. Then length = 3(10) - 4 = 30 - 4 = 26 feet. The dimensions are 10 feet by 26 feet.

6. Noah is designing a rectangular playground on a coordinate grid. One corner is at the origin (0,0), and the playground extends along the positive x-axis and positive y-axis. The length along the x-axis is 7 meters more than the width along the y-axis. The area of the playground is 120 square meters. Write a quadratic equation in standard form that represents this situation, then solve for the width by factoring. Answer: 8 Solution: Let w represent the width of the playground (in meters). Since the length is 7 meters more than the width, the length is w + 7. The area of a rectangle is length times width.
   Full step-by-step solution

   Step 1: Let w represent the width of the playground (in meters). Since the length is 7 meters more than the width, the length is w + 7. Step 2: The area of a rectangle is length times width. So the area is (w + 7) * w = w^2 + 7w. Step 3: Set this equal to the given area of 120: w^2 + 7w = 120. Step 4: Subtract 120 from both sides to get standard form: w^2 + 7w - 120 = 0. Step 5: Factor the quadratic. We need two numbers that multiply to -120 and add to 7. These numbers are 15 and -8 because 15 * (-8) = -120 and 15 + (-8) = 7. Step 6: Write the factored form: (w + 15)(w - 8) = 0. Step 7: Apply the zero product property: w + 15 = 0 or w - 8 = 0. Step 8: Solve each equation: w = -15 or w = 8. Step 9: Since width cannot be negative, discard w = -15. The width is 8 meters. The answer is 8.

7. x² + 5x = 0 Answer: x = 0, -5 Solution: Factor the left side of the equation: x(x + 5) = 0 Apply the zero product property: If the product of two factors is zero, then at least one of the factors must be zero.
   Full step-by-step solution

   Step 1: Factor the left side of the equation: x(x + 5) = 0 Step 2: Apply the zero product property: If the product of two factors is zero, then at least one of the factors must be zero. Step 3: Set each factor equal to zero and solve: x = 0 x + 5 = 0 → x = -5 Step 4: The solutions are x = 0 and x = -5.