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Factor Trinomials

Grade 9 · Algebra · Worksheet 2

  1. A physics class is designing a projectile motion experiment where the height of a ball above ground is modeled by the quadratic equation h(t) = 3t² + 11t + 6, where h is height in meters and t is time in seconds. The teacher asks students to factor this expression to determine at what times the ball will be at ground level. Using the AC method, what is the factored form of this quadratic expression? Answer: ______________
  2. A rectangular garden has an area represented by the quadratic expression 2x² + 7x + 6 square meters, where x represents the width in meters. The length of the garden is 5 meters more than its width. Factor the quadratic expression to determine the binomial expressions that represent the garden's dimensions. Answer: ______________
  3. Charlotte is designing a rectangular garden for a school project. The area of the garden (in square meters) is represented by the quadratic expression 8x² + 26x + 15, where x is a positive integer representing the width in meters. The length and width of the garden are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression to find the binomial expressions that represent the garden's dimensions. Answer: ______________
  4. A rectangular garden is designed with an area represented by the quadratic expression 2x² + 11x + 12 square meters. The garden is visually divided into four smaller rectangles using the area model, where the length and width are binomial expressions with integer coefficients. What is the factored form that represents the dimensions of this garden? Answer: ______________
  5. Mere is designing a rectangular pathway for a school project. The area of the pathway in square meters is given by the quadratic expression 6x² + 25x + 14, where x is a positive integer representing the width in meters. The length and width of the pathway are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression completely to find the binomial expressions that represent the possible dimensions of the pathway. Answer: ______________
  6. Isabella is designing a rectangular community garden. The area of the garden, in square meters, is given by the quadratic expression 6x² + 29x + 35, where x represents a positive integer. To determine the possible length and width of the garden (in meters), Isabella needs to factor this trinomial using the AC method. What is the factored form of 6x² + 29x + 35? Answer: ______________
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Answer Key & Explanations

Factor Trinomials · Grade 9 · Worksheet 2

  1. A physics class is designing a projectile motion experiment where the height of a ball above ground is modeled by the quadratic equation h(t) = 3t² + 11t + 6, where h is height in meters and t is time in seconds. The teacher asks students to factor this expression to determine at what times the ball will be at ground level. Using the AC method, what is the factored form of this quadratic expression? Answer: (3t + 2)(t + 3) Solution: Identify the coefficients: a = 3, b = 11, c = 6 Multiply a and c: 3 × 6 = 18 Find two numbers that multiply to 18 and add to 11: 2 and 9 (since 2 × 9 = 18 and 2 + 9 = 11) Rewrite the middle term using these numbers: 3t² + 2t + 9t + 6 Factor by grouping: (3t² + 2t) + (9t + 6) = t(3t + 2) + 3(3t +…
    Full step-by-step solution

    Step 1: Identify the coefficients: a = 3, b = 11, c = 6 Step 2: Multiply a and c: 3 × 6 = 18 Step 3: Find two numbers that multiply to 18 and add to 11: 2 and 9 (since 2 × 9 = 18 and 2 + 9 = 11) Step 4: Rewrite the middle term using these numbers: 3t² + 2t + 9t + 6 Step 5: Factor by grouping: (3t² + 2t) + (9t + 6) = t(3t + 2) + 3(3t + 2) Step 6: Factor out the common binomial: (3t + 2)(t + 3) The factored form is (3t + 2)(t + 3).

  2. A rectangular garden has an area represented by the quadratic expression 2x² + 7x + 6 square meters, where x represents the width in meters. The length of the garden is 5 meters more than its width. Factor the quadratic expression to determine the binomial expressions that represent the garden's dimensions. Answer: (2x+3)(x+2) Solution: The AC method for factoring quadratic trinomials involves finding two numbers that multiply to A×C and add to B, then using these numbers to split the middle term and factor by grouping.
    Full step-by-step solution

    The AC method for factoring quadratic trinomials involves finding two numbers that multiply to A×C and add to B, then using these numbers to split the middle term and factor by grouping. This technique is particularly useful for factoring expressions that model real-world situations like area calculations, where the factors represent dimensions of geometric shapes.

  3. Charlotte is designing a rectangular garden for a school project. The area of the garden (in square meters) is represented by the quadratic expression 8x² + 26x + 15, where x is a positive integer representing the width in meters. The length and width of the garden are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression to find the binomial expressions that represent the garden's dimensions. Answer: (4x + 3)(2x + 5) Solution: Identify coefficients: a = 8, b = 26, c = 15 Calculate a × 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) Rewrite the middle term using these numbers: 8x² + 20x + 6x + 15 Group terms: (8x² + 20x) + (6x + 15) Factor each group:…
    Full step-by-step solution

    Step 1: Identify coefficients: a = 8, b = 26, c = 15 Step 2: Calculate a × 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 using these numbers: 8x² + 20x + 6x + 15 Step 5: Group terms: (8x² + 20x) + (6x + 15) Step 6: Factor each group: 4x(2x + 5) + 3(2x + 5) Step 7: Factor out the common binomial: (4x + 3)(2x + 5) The factored form is (4x + 3)(2x + 5).

  4. A rectangular garden is designed with an area represented by the quadratic expression 2x² + 11x + 12 square meters. The garden is visually divided into four smaller rectangles using the area model, where the length and width are binomial expressions with integer coefficients. What is the factored form that represents the dimensions of this garden? Answer: (2x+3)(x+4) Solution: Identify a, b, and c in the quadratic: a = 2, b = 11, c = 12 Multiply a and c: 2 × 12 = 24 Find two numbers that multiply to 24 and add to 11: 3 and 8 (since 3 × 8 = 24 and 3 + 8 = 11) Rewrite the middle term using these numbers: 2x² + 3x + 8x + 12 Factor by grouping: (2x² + 3x) + (8x + 12) =…
    Full step-by-step solution

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

  5. Mere is designing a rectangular pathway for a school project. The area of the pathway in square meters is given by the quadratic expression 6x² + 25x + 14, where x is a positive integer representing the width in meters. The length and width of the pathway are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression completely to find the binomial expressions that represent the possible dimensions of the pathway. Answer: (3x + 2)(2x + 7) Solution: Identify the coefficients: a = 6, b = 25, c = 14. Compute a × c = 6 × 14 = 84. Find two numbers that multiply to 84 and add to 25.
    Full step-by-step solution

    Step 1: Identify the coefficients: a = 6, b = 25, c = 14. Step 2: Compute a × c = 6 × 14 = 84. Step 3: Find two numbers that multiply to 84 and add to 25. The numbers are 21 and 4 because 21 × 4 = 84 and 21 + 4 = 25. Step 4: Rewrite the middle term 25x as 21x + 4x: 6x² + 21x + 4x + 14. Step 5: Group the terms: (6x² + 21x) + (4x + 14). Step 6: Factor each group: 3x(2x + 7) + 2(2x + 7). Step 7: Factor out the common binomial (2x + 7): (3x + 2)(2x + 7). The factored form is (3x + 2)(2x + 7).

  6. Isabella is designing a rectangular community garden. The area of the garden, in square meters, is given by the quadratic expression 6x² + 29x + 35, where x represents a positive integer. To determine the possible length and width of the garden (in meters), Isabella needs to factor this trinomial using the AC method. What is the factored form of 6x² + 29x + 35? Answer: (3x + 7)(2x + 5) Solution: Identify coefficients: a = 6, b = 29, c = 35. Compute a*c = 6 * 35 = 210. Find two numbers that multiply to 210 and add to 29.
    Full step-by-step solution

    Step 1: Identify coefficients: a = 6, b = 29, c = 35. Step 2: Compute a*c = 6 * 35 = 210. Step 3: Find two numbers that multiply to 210 and add to 29. The numbers are 14 and 15 because 14 * 15 = 210 and 14 + 15 = 29. Step 4: Rewrite the middle term 29x as 14x + 15x: 6x² + 14x + 15x + 35. Step 5: Group the terms: (6x² + 14x) + (15x + 35). Step 6: Factor each group: 2x(3x + 7) + 5(3x + 7). Step 7: Factor out the common binomial (3x + 7): (3x + 7)(2x + 5). The factored form is (3x + 7)(2x + 5).