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Factor Quadratics

Grade 9 · Algebra · Worksheet 2

  1. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the function h(t) = -16t² + 64t + 80, where h is the height in feet and t is the time in seconds. The teacher asks students to factor this quadratic expression to determine at what time the ball will hit the ground. What is the factored form of the height function? Answer: ______________
  2. 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: ______________
  3. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length of the garden is (2x + 3) meters, what expression represents the width of the garden in meters? Answer: ______________
  4. Factor completely: 10x² - 37x + 30 = ? Answer: ______________
  5. 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: ______________
  6. Factor completely: 8x² - 22x + 15 = ? Answer: ______________
  7. A physics class is designing a projectile launcher that follows the path h(t) = -16t² + 64t + 80, where h is the height in feet and t is time in seconds. The teacher asks students to factor this quadratic expression to determine at what times the projectile will be at ground level. Factor the expression completely to find these times. Answer: ______________
  8. 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: ______________
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Answer Key & Explanations

Factor Quadratics · Grade 9 · Worksheet 2

  1. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the function h(t) = -16t² + 64t + 80, where h is the height in feet and t is the time in seconds. The teacher asks students to factor this quadratic expression to determine at what time the ball will hit the ground. What is the factored form of the height function? Answer: -16(t - 5)(t + 1) Solution: Start with the quadratic function: h(t) = -16t² + 64t + 80 Factor out the greatest common factor, which is -16: h(t) = -16(t² - 4t - 5) Now factor the quadratic inside the parentheses: t² - 4t - 5 Find two numbers that multiply to -5 and add to -4: -5 and 1 Write the factored form: (t - 5)(t +…
    Full step-by-step solution

    Step 1: Start with the quadratic function: h(t) = -16t² + 64t + 80 Step 2: Factor out the greatest common factor, which is -16: h(t) = -16(t² - 4t - 5) Step 3: Now factor the quadratic inside the parentheses: t² - 4t - 5 Step 4: Find two numbers that multiply to -5 and add to -4: -5 and 1 Step 5: Write the factored form: (t - 5)(t + 1) Step 6: Include the -16 we factored out: h(t) = -16(t - 5)(t + 1) The factored form is -16(t - 5)(t + 1).

  2. 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.

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

    We are given: Area = 2x² + 7x + 6 Length = (2x + 3) Width = ? Step 1: Area = Length × Width So, Width = Area ÷ Length Width = (2x² + 7x + 6) ÷ (2x + 3) Step 2: Factor the quadratic 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) Step 5: Factor each group: x(2x + 3) + 2(2x + 3) Step 6: Factor out (2x + 3): (2x + 3)(x + 2) Step 7: So Area = (2x + 3)(x + 2) Length = (2x + 3) Therefore Width = (x + 2) Final answer: x + 2

  4. Factor completely: 10x² - 37x + 30 = ? Answer: (5x - 6)(2x - 5) Solution: Multiply the leading coefficient (10) by the constant term (30): 10 × 30 = 300. Find two numbers that multiply to 300 and add to -37. Since the product is positive and the sum is negative, both numbers are negative.
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (10) by the constant term (30): 10 × 30 = 300. Step 2: Find two numbers that multiply to 300 and add to -37. Since the product is positive and the sum is negative, both numbers are negative. The numbers are -12 and -25 because (-12) × (-25) = 300 and (-12) + (-25) = -37. Step 3: Rewrite the middle term -37x as -12x - 25x: 10x² - 12x - 25x + 30. Step 4: Group the terms: (10x² - 12x) + (-25x + 30). Step 5: Factor out the greatest common factor from each group: 2x(5x - 6) - 5(5x - 6). Step 6: Factor out the common binomial (5x - 6): (5x - 6)(2x - 5). The completely factored form is (5x - 6)(2x - 5).

  5. 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.

  6. Factor completely: 8x² - 22x + 15 = ? Answer: (4x - 5)(2x - 3) Solution: Multiply the leading coefficient (8) by the constant term (15): 8 × 15 = 120. Find two numbers that multiply to 120 and add to -22: -10 and -12. Rewrite the middle term using these numbers: 8x² - 10x - 12x + 15.
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (8) by the constant term (15): 8 × 15 = 120. Step 2: Find two numbers that multiply to 120 and add to -22: -10 and -12. Step 3: Rewrite the middle term using these numbers: 8x² - 10x - 12x + 15. Step 4: Factor by grouping: (8x² - 10x) + (-12x + 15). Step 5: Factor out common factors: 2x(4x - 5) - 3(4x - 5). Step 6: Factor out the common binomial: (4x - 5)(2x - 3). The completely factored form is (4x - 5)(2x - 3).

  7. A physics class is designing a projectile launcher that follows the path h(t) = -16t² + 64t + 80, where h is the height in feet and t is time in seconds. The teacher asks students to factor this quadratic expression to determine at what times the projectile will be at ground level. Factor the expression completely to find these times. Answer: t = 5 and t = -1 Solution: In projectile motion problems, factoring quadratic expressions helps determine key events like when an object reaches ground level.
    Full step-by-step solution

    In projectile motion problems, factoring quadratic expressions helps determine key events like when an object reaches ground level. The factored form reveals the roots of the equation, which correspond to specific times in the object's trajectory. For different quadratic models, you would follow similar factoring procedures to extract meaningful information about the physical situation.

  8. 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.