Factoring Quadratic Expressions

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Answer Key & Explanations

Factoring Quadratic Expressions · Grade 10 · Worksheet 1

1. 3x² - 10x - 8 = 0 Answer: x = 4, -2/3 Solution: Multiply the leading coefficient and constant term: 3 × (-8) = -24 Find two numbers that multiply to -24 and add to -10: -12 and 2 Rewrite the middle term: 3x² - 12x + 2x - 8 = 0 Factor by grouping: (3x² - 12x) + (2x - 8) = 3x(x - 4) + 2(x - 4) = 0 Factor out the common binomial: (3x + 2)(x - 4)…
   Full step-by-step solution

   Step 1: Multiply the leading coefficient and constant term: 3 × (-8) = -24 Step 2: Find two numbers that multiply to -24 and add to -10: -12 and 2 Step 3: Rewrite the middle term: 3x² - 12x + 2x - 8 = 0 Step 4: Factor by grouping: (3x² - 12x) + (2x - 8) = 3x(x - 4) + 2(x - 4) = 0 Step 5: Factor out the common binomial: (3x + 2)(x - 4) = 0 Step 6: Set each factor equal to zero: 3x + 2 = 0 or x - 4 = 0 Step 7: Solve each equation: x = -2/3 or x = 4 The solutions are x = 4 and x = -2/3.

2. A physics class is designing a parabolic water fountain where the water's path follows the equation h(t) = -5t² + 20t + 1, where h is the height in meters and t is time in seconds. The students want to know at what times the water will reach a height of 16 meters. Solve the equation -5t² + 20t + 1 = 16 to find these times. Answer: 1 and 3 Solution: In projectile motion problems, the height of an object is often modeled by a quadratic equation.
   Full step-by-step solution

   In projectile motion problems, the height of an object is often modeled by a quadratic equation. To find when the object reaches a specific height, you set the equation equal to that height and solve the resulting quadratic. This typically involves moving all terms to one side to set the equation to zero, then factoring or using the quadratic formula to find the time values.

3. Factor: 8x² + 26x + 15 Answer: (2x + 5)(4x + 3) Solution: Identify a = 8, b = 26, c = 15. Multiply a and c: 8 × 15 = 120. Find two numbers that multiply to 120 and add to 26: 20 and 6 (since 20 × 6 = 120 and 20 + 6 = 26).
   Full step-by-step solution

   Step 1: Identify a = 8, b = 26, c = 15. Step 2: Multiply a and c: 8 × 15 = 120. Step 3: Find two numbers that multiply to 120 and add to 26: 20 and 6 (since 20 × 6 = 120 and 20 + 6 = 26). Step 4: Rewrite the middle term: 8x² + 20x + 6x + 15. Step 5: Factor by grouping: (8x² + 20x) + (6x + 15). Step 6: Factor out the GCF from each group: 4x(2x + 5) + 3(2x + 5). Step 7: Factor out the common binomial (2x + 5): (2x + 5)(4x + 3). The factored form is (2x + 5)(4x + 3).

4. Charlotte is designing a rectangular solar panel for a school project. The area of the panel is given by the polynomial 6x² + 7x – 5 square feet. If the length of the panel is (2x – 1) feet, what expression represents the width of the panel in terms of x? Answer: 3x + 5 Solution: Area of a rectangle = length × width, so width = area / length. The area is 6x² + 7x – 5 and the length is 2x – 1. We need to factor 6x² + 7x – 5.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so width = area / length. Step 2: The area is 6x² + 7x – 5 and the length is 2x – 1. Step 3: We need to factor 6x² + 7x – 5. Find two numbers that multiply to (6)(–5) = –30 and add to 7. These numbers are 10 and –3. Step 4: Rewrite the middle term: 6x² + 10x – 3x – 5. Step 5: Factor by grouping: 2x(3x + 5) – 1(3x + 5) = (2x – 1)(3x + 5). Step 6: Therefore, width = (6x² + 7x – 5) / (2x – 1) = 3x + 5. The answer is 3x + 5.

5. Mere is designing a rectangular metal sheet for a school art project. The area of the sheet (in square centimeters) is given by the quadratic expression 6x² + 23x - 18. If the length of the sheet is (3x - 2) centimeters, what expression represents the width of the sheet in centimeters? Answer: 2x + 9 Solution: Area of a rectangle = length × width, so width = area / length. Area = 6x² + 23x - 18, length = (3x - 2). To find the width, divide the area by the length: (6x² + 23x - 18) / (3x - 2).
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so width = area / length. Step 2: Area = 6x² + 23x - 18, length = (3x - 2). Step 3: To find the width, divide the area by the length: (6x² + 23x - 18) / (3x - 2). Step 4: Perform polynomial division or factor the quadratic. To factor 6x² + 23x - 18, look for two binomials whose product gives this expression. Step 5: Multiply (3x - 2)(2x + 9) = 3x(2x) + 3x(9) - 2(2x) - 2(9) = 6x² + 27x - 4x - 18 = 6x² + 23x - 18. Step 6: Since the product equals the area, the width is the other factor: (2x + 9). The width of the sheet is 2x + 9 centimeters.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of the triangle is 16 square units. Find the value of x. Answer: 4 Solution: The vertices are at (0,0), (x,0), and (0,2x). This means one leg of the right triangle is along the x-axis from (0,0) to (x,0), so its length is x.
   Full step-by-step solution

   Step 1: Understand the triangle's vertices. The vertices are at (0,0), (x,0), and (0,2x). This means one leg of the right triangle is along the x-axis from (0,0) to (x,0), so its length is x. The other leg is along the y-axis from (0,0) to (0,2x), so its length is 2x. Step 2: Identify the base and height. Since it's a right triangle with legs along the axes, we can take the leg along the x-axis as the base and the leg along the y-axis as the height. Base = x Height = 2x Step 3: Write the formula for the area of a triangle. Area = (1/2) * base * height Step 4: Substitute the known area and the expressions for base and height. Area = 16 So: 16 = (1/2) * (x) * (2x) Step 5: Simplify the equation. 16 = (1/2) * 2x * x 16 = (1/2) * 2x^2 16 = x^2 Step 6: Solve for x. x^2 = 16 x = 4 or x = -4 Step 7: Interpret the solution. Since x is a length (distance along the x-axis), it must be positive. Therefore, x = 4. Final answer: x = 4

7. Factor: 8x² + 14x - 15 = ? Answer: (2x + 5)(4x - 3) Solution: Identify a = 8, b = 14, c = -15. Multiply a and c: 8 × (-15) = -120. Find two numbers that multiply to -120 and add to 14.
   Full step-by-step solution

   Step 1: Identify a = 8, b = 14, c = -15. Step 2: Multiply a and c: 8 × (-15) = -120. Step 3: Find two numbers that multiply to -120 and add to 14. The numbers are 20 and -6 because 20 × (-6) = -120 and 20 + (-6) = 14. Step 4: Rewrite the middle term using these numbers: 8x² + 20x - 6x - 15. Step 5: Group the terms: (8x² + 20x) + (-6x - 15). Step 6: Factor out the GCF from each group: 4x(2x + 5) - 3(2x + 5). Step 7: Factor out the common binomial (2x + 5): (2x + 5)(4x - 3). The factored form is (2x + 5)(4x - 3).

8. Sophia is designing a rectangular rose garden for her school's landscaping project. The area of the garden is given by the quadratic expression 6x² + 31x + 35 square feet, where x is a positive integer representing a scaling factor. The length of the garden is known to be (3x + 5) feet. What expression represents the width of the garden in feet? Answer: 2x + 7 Solution: Area of a rectangle = length × width, so width = area ÷ length. Area = 6x² + 31x + 35, length = 3x + 5. To find width, factor the quadratic 6x² + 31x + 35.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so width = area ÷ length. Step 2: Area = 6x² + 31x + 35, length = 3x + 5. Step 3: To find width, factor the quadratic 6x² + 31x + 35. Step 4: Look for two numbers that multiply to 6 × 35 = 210 and add to 31. The numbers are 10 and 21. Step 5: Rewrite the middle term: 6x² + 10x + 21x + 35. Step 6: Group the terms: (6x² + 10x) + (21x + 35). Step 7: Factor each group: 2x(3x + 5) + 7(3x + 5). Step 8: Factor out the common binomial: (3x + 5)(2x + 7). Step 9: Since length is (3x + 5), width must be (2x + 7). The answer is 2x + 7.