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Factor Quadratics

Grade 9 · Algebra · Worksheet 1

  1. 2x² - 11x + 12 = ? Answer: ______________
  2. A local theater is designing a new stage backdrop that will have a rectangular opening for special effects. The area of the opening is given by the quadratic expression 3x² + 10x + 8 square feet, where x represents the additional length beyond the minimum dimensions. The theater director knows that the width of the opening is (x + 2) feet. What expression represents the length of the opening in factored form? Answer: ______________
  3. A rectangular garden has an area that can be expressed as 2x² + 7x + 6 square meters. If the length of the garden is (2x + 3) meters, what is the width of the garden in terms of x? Answer: ______________
  4. A physics student is designing a parabolic arch for a model bridge. The arch's height above ground is modeled by the function h(x) = -2x² + 12x - 16, where x represents the horizontal distance from the left support in meters. Factor this quadratic expression to determine at what horizontal distances from the left support the arch touches the ground. Answer: ______________
  5. A physics class is designing a projectile launcher that follows the path h(t) = -16t² + 64t + 5, where h is the height in feet and t is time in seconds. To analyze the maximum height, they need to rewrite the equation in vertex form by factoring. What is the factored form of the quadratic expression -16t² + 64t + 5? Answer: ______________
  6. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 16x - 24. The company breaks even when profit equals zero. Factor this quadratic expression to determine the number of units where the company breaks even. Answer: ______________
  7. A robotics team is designing a projectile launcher for a competition. The height h (in meters) of their projectile after t seconds is modeled by the function h(t) = -5t² + 20t + 15. To determine the maximum height the projectile reaches, they need to rewrite this quadratic function in vertex form by factoring. What is the vertex form of h(t) = -5t² + 20t + 15? Answer: ______________
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Answer Key & Explanations

Factor Quadratics · Grade 9 · Worksheet 1

  1. 2x² - 11x + 12 = ? Answer: (2x - 3)(x - 4) Solution: Identify a = 2, b = -11, c = 12 Multiply a × c = 2 × 12 = 24 Find two numbers that multiply to 24 and add to -11: -3 and -8 Rewrite the middle term: 2x² - 3x - 8x + 12 Factor by grouping: x(2x - 3) - 4(2x - 3) Factor out the common binomial: (2x - 3)(x - 4) The answer is (2x - 3)(x - 4).
    Full step-by-step solution

    Step 1: Identify a = 2, b = -11, c = 12 Step 2: Multiply a × c = 2 × 12 = 24 Step 3: Find two numbers that multiply to 24 and add to -11: -3 and -8 Step 4: Rewrite the middle term: 2x² - 3x - 8x + 12 Step 5: Factor by grouping: x(2x - 3) - 4(2x - 3) Step 6: Factor out the common binomial: (2x - 3)(x - 4) The answer is (2x - 3)(x - 4).

  2. A local theater is designing a new stage backdrop that will have a rectangular opening for special effects. The area of the opening is given by the quadratic expression 3x² + 10x + 8 square feet, where x represents the additional length beyond the minimum dimensions. The theater director knows that the width of the opening is (x + 2) feet. What expression represents the length of the opening in factored form? Answer: (3x + 4) Solution: In geometry, the area of a rectangle equals length times width. When working with polynomial expressions for area, if one dimension is known, the other can be found by factoring or dividing the area expression.
    Full step-by-step solution

    In geometry, the area of a rectangle equals length times width. When working with polynomial expressions for area, if one dimension is known, the other can be found by factoring or dividing the area expression. This concept applies to various real-world situations like calculating dimensions of rooms, gardens, or architectural elements when the total area and one measurement are known.

  3. A rectangular garden has an area that can be expressed as 2x² + 7x + 6 square meters. If the length of the garden is (2x + 3) meters, what is the width of the garden in terms of x? Answer: (x + 2) Solution: Area = 2x² + 7x + 6 Length = (2x + 3) Width = ? Write the relationship between area, length, and width.
    Full step-by-step solution

    We are given: Area = 2x² + 7x + 6 Length = (2x + 3) Width = ? Step 1: Write the relationship between area, length, and width. Area = Length × Width So, Width = Area ÷ Length Width = (2x² + 7x + 6) ÷ (2x + 3) Step 2: 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 the middle term using 3 and 4: 2x² + 7x + 6 = 2x² + 3x + 4x + 6 Step 4: Factor by grouping: Group (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 substitute back into the width formula: Width = (2x² + 7x + 6) ÷ (2x + 3) = [(2x + 3)(x + 2)] ÷ (2x + 3) Step 7: Cancel the common factor (2x + 3) (assuming 2x + 3 ≠ 0): Width = x + 2 Final answer: The width is (x + 2) meters.

  4. A physics student is designing a parabolic arch for a model bridge. The arch's height above ground is modeled by the function h(x) = -2x² + 12x - 16, where x represents the horizontal distance from the left support in meters. Factor this quadratic expression to determine at what horizontal distances from the left support the arch touches the ground. Answer: 2 and 4 Solution: Set up the equation for when the arch touches the ground: -2x² + 12x - 16 = 0 Factor out the common factor of -2: -2(x² - 6x + 8) = 0 Factor the quadratic inside the parentheses: x² - 6x + 8 = (x - 2)(x - 4) Write the complete factored form: -2(x - 2)(x - 4) = 0 Set each factor equal to zero: x…
    Full step-by-step solution

    Step 1: Set up the equation for when the arch touches the ground: -2x² + 12x - 16 = 0 Step 2: Factor out the common factor of -2: -2(x² - 6x + 8) = 0 Step 3: Factor the quadratic inside the parentheses: x² - 6x + 8 = (x - 2)(x - 4) Step 4: Write the complete factored form: -2(x - 2)(x - 4) = 0 Step 5: Set each factor equal to zero: x - 2 = 0 or x - 4 = 0 Step 6: Solve for x: x = 2 or x = 4 Step 7: Interpret the results: The arch touches the ground at 2 meters and 4 meters from the left support. The answer is 2 and 4.

  5. A physics class is designing a projectile launcher that follows the path h(t) = -16t² + 64t + 5, where h is the height in feet and t is time in seconds. To analyze the maximum height, they need to rewrite the equation in vertex form by factoring. What is the factored form of the quadratic expression -16t² + 64t + 5? Answer: -16(t - 2)² + 69 Solution: Start with the quadratic expression: -16t² + 64t + 5 Factor out -16 from the first two terms: -16(t² - 4t) + 5 Complete the square inside the parentheses: t² - 4t becomes (t² - 4t + 4) - 4 = (t - 2)² - 4 Substitute back: -16[(t - 2)² - 4] + 5 Distribute the -16: -16(t - 2)² + 64 + 5 Combine…
    Full step-by-step solution

    Step 1: Start with the quadratic expression: -16t² + 64t + 5 Step 2: Factor out -16 from the first two terms: -16(t² - 4t) + 5 Step 3: Complete the square inside the parentheses: t² - 4t becomes (t² - 4t + 4) - 4 = (t - 2)² - 4 Step 4: Substitute back: -16[(t - 2)² - 4] + 5 Step 5: Distribute the -16: -16(t - 2)² + 64 + 5 Step 6: Combine constants: -16(t - 2)² + 69 The answer is -16(t - 2)² + 69.

  6. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 16x - 24. The company breaks even when profit equals zero. Factor this quadratic expression to determine the number of units where the company breaks even. Answer: 2 and 6 Solution: Set up the equation for break-even point: -2x² + 16x - 24 = 0 Factor out the greatest common factor: -2(x² - 8x + 12) = 0 Factor the quadratic inside the parentheses: x² - 8x + 12 = (x - 2)(x - 6) Write the complete factored form: -2(x - 2)(x - 6) = 0 Set each factor equal to zero: x - 2 = 0 or…
    Full step-by-step solution

    Step 1: Set up the equation for break-even point: -2x² + 16x - 24 = 0 Step 2: Factor out the greatest common factor: -2(x² - 8x + 12) = 0 Step 3: Factor the quadratic inside the parentheses: x² - 8x + 12 = (x - 2)(x - 6) Step 4: Write the complete factored form: -2(x - 2)(x - 6) = 0 Step 5: Set each factor equal to zero: x - 2 = 0 or x - 6 = 0 Step 6: Solve for x: x = 2 or x = 6 The company breaks even when selling 2 units or 6 units.

  7. A robotics team is designing a projectile launcher for a competition. The height h (in meters) of their projectile after t seconds is modeled by the function h(t) = -5t² + 20t + 15. To determine the maximum height the projectile reaches, they need to rewrite this quadratic function in vertex form by factoring. What is the vertex form of h(t) = -5t² + 20t + 15? Answer: -5(t - 2)² + 35 Solution: Start with the quadratic function: h(t) = -5t² + 20t + 15 Factor -5 from the first two terms: h(t) = -5(t² - 4t) + 15 Complete the square inside the parentheses: Take half of -4 (which is -2), square it to get 4 Add and subtract 4 inside the parentheses: h(t) = -5(t² - 4t + 4 - 4) + 15 Rewrite…
    Full step-by-step solution

    Step 1: Start with the quadratic function: h(t) = -5t² + 20t + 15 Step 2: Factor -5 from the first two terms: h(t) = -5(t² - 4t) + 15 Step 3: Complete the square inside the parentheses: Take half of -4 (which is -2), square it to get 4 Step 4: Add and subtract 4 inside the parentheses: h(t) = -5(t² - 4t + 4 - 4) + 15 Step 5: Rewrite as: h(t) = -5((t - 2)² - 4) + 15 Step 6: Distribute the -5: h(t) = -5(t - 2)² + 20 + 15 Step 7: Combine constants: h(t) = -5(t - 2)² + 35 The vertex form is -5(t - 2)² + 35.