Quadratic by Factoring

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Answer Key & Explanations

Quadratic by Factoring · Grade 9 · Worksheet 2

1. 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.

2. x² - 14x + 48 = 0 Answer: x = 6, x = 8 Solution: Factor the quadratic x² - 14x + 48 = 0. Find two numbers that multiply to 48 and add to -14. The numbers are -6 and -8 because (-6) × (-8) = 48 and (-6) + (-8) = -14.
   Full step-by-step solution

   Step 1: Factor the quadratic x² - 14x + 48 = 0. Step 2: Find two numbers that multiply to 48 and add to -14. The numbers are -6 and -8 because (-6) × (-8) = 48 and (-6) + (-8) = -14. Step 3: Write the factored form: (x - 6)(x - 8) = 0. Step 4: Apply the zero product property: x - 6 = 0 or x - 8 = 0. Step 5: Solve each equation: x = 6 or x = 8. The solutions are x = 6 and x = 8.

3. Noah is designing a rectangular skateboard ramp. The area of the ramp's surface is 66 square feet. The length of the ramp is 5 feet more than its width. What are the dimensions of the ramp in feet? Answer: Width = 6 feet, Length = 11 feet Solution: Let w represent the width of the ramp in feet. Then the length is w + 5 feet. Area = length × width, so w(w + 5) = 66.
   Full step-by-step solution

   Let w represent the width of the ramp in feet. Then the length is w + 5 feet. Area = length × width, so w(w + 5) = 66. Expand: w² + 5w = 66. Subtract 66 from both sides: w² + 5w - 66 = 0. Factor the quadratic: (w + 11)(w - 6) = 0. Apply the zero product property: w + 11 = 0 or w - 6 = 0, so w = -11 or w = 6. Since width cannot be negative, w = 6 feet. Then length = w + 5 = 11 feet. The dimensions are 6 feet by 11 feet.

4. 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.

5. 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 a quadratic equation in standard form that represents this situation and solve for x by factoring. Answer: x=4 Solution: The vertices are (0,0), (x,0), and (0,6). This means the triangle has legs along the x-axis and y-axis. The base is along the x-axis from (0,0) to (x,0), so base length = |x|.
   Full step-by-step solution

   Step 1: Understand the triangle The vertices are (0,0), (x,0), and (0,6). This means the triangle has legs along the x-axis and y-axis. 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 Area of a triangle = (1/2) * base * height Given area = 12, base = x, height = 6: (1/2) * x * 6 = 12 Step 3: Simplify the equation (1/2) * 6 = 3, so: 3 * x = 12 Divide both sides by 3: x = 4 Step 4: Check if a quadratic equation is needed The problem says: "Write a quadratic equation in standard form that represents this situation and solve for x by factoring." But from the area formula, we got a linear equation, not quadratic. This means we likely need to consider that the base length is |x|, so maybe x could be negative, but area uses positive lengths, so base = |x|. But if x is negative, the point (x,0) is left of the origin, but the triangle's vertices (0,0), (x,0), (0,6) still form a right triangle with legs |x| and 6. Area = (1/2) * |x| * 6 = 12 So |x| * 3 = 12 |x| = 4 So x = 4 or x = -4. Step 5: Make a quadratic equation From |x| = 4, square both sides: x^2 = 16 So x^2 - 16 = 0 is the quadratic in standard form. Step 6: Solve by factoring x^2 - 16 = 0 This is difference of squares: (x - 4)(x + 4) = 0 So x - 4 = 0 or x + 4 = 0 Thus x = 4 or x = -4. Step 7: Interpret the answer Both x = 4 and x = -4 give area 12, but the problem likely expects the positive solution x = 4 as the final answer. Final: The quadratic equation is x^2 - 16 = 0, and solving gives x = 4 or x = -4, with x = 4 being the expected answer.

6. A technology company is designing a new smartphone with a rectangular screen. The screen's area is 216 square centimeters. The length of the screen is 6 centimeters more than its width. To optimize manufacturing costs, engineers need to determine the exact dimensions of the screen. What are the width and length of the smartphone screen in centimeters? Answer: 12 and 18 Solution: The mathematical approach involves setting up a quadratic equation where the product of length and width equals the area.
   Full step-by-step solution

   Many real-world design problems involve finding dimensions when given area and a relationship between length and width. The mathematical approach involves setting up a quadratic equation where the product of length and width equals the area. Factoring such equations reveals the possible dimensions, though only positive measurements make sense in physical contexts.

7. A tech startup is designing a new smartphone with a rectangular screen. The screen's area is 96 square inches, and the length of the screen is 4 inches 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 inches? Answer: 8 and 12 Solution: The key is to represent the unknown width as a variable, express the length in terms of that variable, set up an area equation, and solve the resulting quadratic.
   Full step-by-step solution

   Many real-world design problems involve finding dimensions when given area and a relationship between length and width. The key is to represent the unknown width as a variable, express the length in terms of that variable, set up an area equation, and solve the resulting quadratic. Factoring works well when the equation can be rewritten as two binomials whose product equals zero, allowing you to find the positive solution that makes sense in the physical context.

8. Mason is designing a rectangular mural for his school's art project. The area of the mural must be 252 square feet. The length of the mural is 4 feet more than three times its width. What are the dimensions of the mural in feet? Answer: width = 7 ft, length = 25 ft Solution: Let w represent the width of the mural in feet. Then the length is 3w + 4 feet.
   Full step-by-step solution

   Let w represent the width of the mural in feet. Then the length is 3w + 4 feet. Area = length × width = (3w + 4)w = 252 So 3w² + 4w = 252 Rewrite as 3w² + 4w - 252 = 0 Factor the quadratic: (3w + 28)(w - 7) = 0 Set each factor equal to zero: 3w + 28 = 0 → w = -28/3 (not valid, width cannot be negative) w - 7 = 0 → w = 7 Width = 7 feet Length = 3(7) + 4 = 21 + 4 = 25 feet The dimensions of the mural are 7 feet by 25 feet.