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Factor Quadratics

Grade 9 · Algebra · Worksheet 2

  1. A construction company is designing a rectangular parking lot with an area represented by the polynomial 3x² + 10x - 8 square meters. The length of the parking lot is given as (x + 4) meters. What expression in factored form represents the width of the parking lot? Answer: ______________
  2. 2x² + 7x + 3 = ? Answer: ______________
  3. Factor completely: 12x² + 37x + 21 = ? Answer: ______________
  4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of this triangle is 16 square units. What is the value of x? Answer: ______________
  5. Sophia is designing a rectangular solar panel for a school project. The area of the solar panel is given by the quadratic expression 6x² + 19x + 10 square centimeters. The length of the panel is (3x + 2) centimeters. What expression in factored form represents the width of the solar panel? Answer: ______________
  6. A company's profit from selling handmade candles follows the quadratic model P(x) = -2x² + 20x + 48, where x represents the price per candle in dollars. The company wants to determine the price points where they break even (profit equals zero). Factor the quadratic expression to find these break-even prices. Answer: ______________
  7. A rectangular garden has an area that can be expressed as x² + 7x + 12 square meters. If the length of the garden is 2 meters longer than the width, find the dimensions of the garden in terms of x. Answer: ______________
  8. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length of the garden is (x + 2) meters, what is the width of the garden in terms of x? Answer: ______________
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Answer Key & Explanations

Factor Quadratics · Grade 9 · Worksheet 2

  1. A construction company is designing a rectangular parking lot with an area represented by the polynomial 3x² + 10x - 8 square meters. The length of the parking lot is given as (x + 4) meters. What expression in factored form represents the width of the parking lot? Answer: (3x - 2) Solution: The area of a rectangle is length × width, so width = area ÷ length Width = (3x² + 10x - 8) ÷ (x + 4) Factor the numerator: 3x² + 10x - 8 Find two numbers that multiply to 3 × (-8) = -24 and add to 10 The numbers are 12 and -2 Rewrite: 3x² + 12x - 2x - 8 Factor by grouping: (3x² + 12x) + (-2x -…
    Full step-by-step solution

    Step 1: The area of a rectangle is length × width, so width = area ÷ length Step 2: Width = (3x² + 10x - 8) ÷ (x + 4) Step 3: Factor the numerator: 3x² + 10x - 8 Step 4: Find two numbers that multiply to 3 × (-8) = -24 and add to 10 Step 5: The numbers are 12 and -2 Step 6: Rewrite: 3x² + 12x - 2x - 8 Step 7: Factor by grouping: (3x² + 12x) + (-2x - 8) Step 8: Factor each group: 3x(x + 4) - 2(x + 4) Step 9: Factor out (x + 4): (x + 4)(3x - 2) Step 10: Width = (x + 4)(3x - 2) ÷ (x + 4) = (3x - 2) The width is (3x - 2) meters.

  2. 2x² + 7x + 3 = ? Answer: (2x + 1)(x + 3) Solution: Identify coefficients: a = 2, b = 7, c = 3 Multiply a × c = 2 × 3 = 6 Find two numbers that multiply to 6 and add to 7: 6 and 1 Rewrite the middle term using these numbers: 2x² + 6x + 1x + 3 Factor by grouping: (2x² + 6x) + (1x + 3) = 2x(x + 3) + 1(x + 3) Factor out the common binomial: (2x +…
    Full step-by-step solution

    Step 1: Identify coefficients: a = 2, b = 7, c = 3 Step 2: Multiply a × c = 2 × 3 = 6 Step 3: Find two numbers that multiply to 6 and add to 7: 6 and 1 Step 4: Rewrite the middle term using these numbers: 2x² + 6x + 1x + 3 Step 5: Factor by grouping: (2x² + 6x) + (1x + 3) = 2x(x + 3) + 1(x + 3) Step 6: Factor out the common binomial: (2x + 1)(x + 3) The answer is (2x + 1)(x + 3).

  3. Factor completely: 12x² + 37x + 21 = ? Answer: (3x + 7)(4x + 3) Solution: Multiply the leading coefficient (12) by the constant term (21): 12 × 21 = 252. Find two numbers that multiply to 252 and add to 37. The numbers are 28 and 9 because 28 × 9 = 252 and 28 + 9 = 37.
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (12) by the constant term (21): 12 × 21 = 252. Step 2: Find two numbers that multiply to 252 and add to 37. The numbers are 28 and 9 because 28 × 9 = 252 and 28 + 9 = 37. Step 3: Rewrite the middle term 37x as 28x + 9x: 12x² + 28x + 9x + 21. Step 4: Factor by grouping: (12x² + 28x) + (9x + 21). Step 5: Factor out the greatest common factor from each group: 4x(3x + 7) + 3(3x + 7). Step 6: Factor out the common binomial (3x + 7): (3x + 7)(4x + 3). The completely factored form is (3x + 7)(4x + 3).

  4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of this triangle is 16 square units. What is the value of x? Answer: 4 Solution: The triangle has vertices at (0,0), (x,0), and (0,2x). This means the base along the x-axis has length x, and the height along the y-axis has length 2x. The area of a right triangle is (1/2) × base × height.
    Full step-by-step solution

    Step 1: The triangle has vertices at (0,0), (x,0), and (0,2x). This means the base along the x-axis has length x, and the height along the y-axis has length 2x. Step 2: The area of a right triangle is (1/2) × base × height. Step 3: Substitute the known values: Area = (1/2) × x × (2x) = (1/2) × 2x² = x². Step 4: Set this equal to the given area: x² = 16. Step 5: Solve for x: x = 4 (since length must be positive). The answer is 4.

  5. Sophia is designing a rectangular solar panel for a school project. The area of the solar panel is given by the quadratic expression 6x² + 19x + 10 square centimeters. The length of the panel is (3x + 2) centimeters. What expression in factored form represents the width of the solar panel? Answer: (2x + 5) Solution: The area of a rectangle is length times width, so width = area / length. Width = (6x² + 19x + 10) / (3x + 2). Factor the numerator 6x² + 19x + 10.
    Full step-by-step solution

    Step 1: The area of a rectangle is length times width, so width = area / length. Step 2: Width = (6x² + 19x + 10) / (3x + 2). Step 3: Factor the numerator 6x² + 19x + 10. Step 4: Find two numbers that multiply to 6 * 10 = 60 and add to 19. The numbers are 15 and 4. Step 5: Rewrite the middle term: 6x² + 15x + 4x + 10. Step 6: Group terms: (6x² + 15x) + (4x + 10). Step 7: Factor each group: 3x(2x + 5) + 2(2x + 5). Step 8: Factor out the common binomial (2x + 5): (2x + 5)(3x + 2). Step 9: Width = (2x + 5)(3x + 2) / (3x + 2) = 2x + 5. The width is (2x + 5) centimeters.

  6. A company's profit from selling handmade candles follows the quadratic model P(x) = -2x² + 20x + 48, where x represents the price per candle in dollars. The company wants to determine the price points where they break even (profit equals zero). Factor the quadratic expression to find these break-even prices. Answer: 12 and -2 Solution: Set up the equation for break-even point: -2x² + 20x + 48 = 0 Factor out -2 from all terms: -2(x² - 10x - 24) = 0 Factor the quadratic inside the parentheses: x² - 10x - 24 = (x - 12)(x + 2) Write the complete factored form: -2(x - 12)(x + 2) = 0 Set each factor equal to zero: x - 12 = 0 gives x…
    Full step-by-step solution

    Step 1: Set up the equation for break-even point: -2x² + 20x + 48 = 0 Step 2: Factor out -2 from all terms: -2(x² - 10x - 24) = 0 Step 3: Factor the quadratic inside the parentheses: x² - 10x - 24 = (x - 12)(x + 2) Step 4: Write the complete factored form: -2(x - 12)(x + 2) = 0 Step 5: Set each factor equal to zero: x - 12 = 0 gives x = 12, and x + 2 = 0 gives x = -2 Step 6: Since price cannot be negative, the meaningful break-even price is $12, but mathematically both solutions are valid. The break-even prices are 12 and -2.

  7. A rectangular garden has an area that can be expressed as x² + 7x + 12 square meters. If the length of the garden is 2 meters longer than the width, find the dimensions of the garden in terms of x. Answer: (x+3) meters by (x+4) meters Solution: The process involves finding two binomials whose product equals the original quadratic.
    Full step-by-step solution

    Factoring quadratic expressions is a fundamental algebraic skill used in many real-world applications like calculating areas, solving projectile motion problems, and optimizing dimensions. The process involves finding two binomials whose product equals the original quadratic. These factors often represent meaningful quantities in applied problems, such as length and width in geometry contexts or time intervals in physics scenarios.

  8. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length of the garden is (x + 2) meters, what is the width of the garden in terms of x? Answer: (2x + 3) Solution: Area = 2x² + 7x + 6 Length = (x + 2) Width = ? Area of rectangle = Length × Width So, Width = Area ÷ Length Width = (2x² + 7x + 6) ÷ (x + 2) We can factor the quadratic expression 2x² + 7x + 6.
    Full step-by-step solution

    We are given: Area = 2x² + 7x + 6 Length = (x + 2) Width = ? Step 1: Area of rectangle = Length × Width So, Width = Area ÷ Length That means: Width = (2x² + 7x + 6) ÷ (x + 2) Step 2: We can factor the quadratic expression 2x² + 7x + 6. We look for two numbers that multiply to 2×6 = 12 and add to 7. Those numbers are 3 and 4. Step 3: Rewrite 7x as 3x + 4x: 2x² + 3x + 4x + 6 Step 4: Group terms: (2x² + 3x) + (4x + 6) Factor each group: x(2x + 3) + 2(2x + 3) Step 5: Factor out (2x + 3): (2x + 3)(x + 2) So: 2x² + 7x + 6 = (2x + 3)(x + 2) Step 6: Now, Width = (2x² + 7x + 6) ÷ (x + 2) = [(2x + 3)(x + 2)] ÷ (x + 2) Cancel (x + 2) (since x ≠ -2 for division to be valid): Width = 2x + 3 Final answer: Width = 2x + 3