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Factor Trinomials

Grade 9 · Algebra · Worksheet 3

  1. A rectangular garden has an area represented by the expression 2x² + 11x + 12 square meters. If the length of the garden is 5 meters more than its width, find the dimensions of the garden by factoring the quadratic expression using the AC method. Answer: ______________
  2. Factor: 14x² + 39x + 10 Answer: ______________
  3. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the equation h(t) = -5t² + 18t + 8, where h is height in meters and t is time in seconds. To analyze when the ball hits the ground, they need to factor this quadratic expression using the AC method. What is the factored form of -5t² + 18t + 8? Answer: ______________
  4. A physics class is designing a catapult that launches projectiles. The height of a projectile above ground is modeled by the quadratic equation h(t) = -16t² + 48t + 64, where h is height in feet and t is time in seconds. To analyze the projectile's flight time, factor this quadratic expression using the AC method to determine when the projectile will return to ground level. Answer: ______________
  5. Factor: 8x² + 26x + 15 = ? Answer: ______________
  6. 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: ______________
  7. Factor: 6x² + 31x + 35 Answer: ______________
  8. Factor: 4x² + 12x + 9 = ? Answer: ______________
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Answer Key & Explanations

Factor Trinomials · Grade 9 · Worksheet 3

  1. A rectangular garden has an area represented by the expression 2x² + 11x + 12 square meters. If the length of the garden is 5 meters more than its width, find the dimensions of the garden by factoring the quadratic expression using the AC method. Answer: (2x + 3)(x + 4) Solution: Area = \( 2x^2 + 11x + 12 \) square meters. Length is 5 meters more than width. We must factor \( 2x^2 + 11x + 12 \) using the AC method.
    Full step-by-step solution

    Let's solve the problem step-by-step. --- **Step 1: Understand the problem** We are given: Area = \( 2x^2 + 11x + 12 \) square meters. Length is 5 meters more than width. We must factor \( 2x^2 + 11x + 12 \) using the AC method. --- **Step 2: Recall the AC method** For a quadratic \( ax^2 + bx + c \): 1. Multiply \( a \times c \). 2. Find two numbers that multiply to \( a \times c \) and add to \( b \). 3. Rewrite the middle term using those two numbers. 4. Factor by grouping. --- **Step 3: Apply AC method** Here: \( a = 2 \), \( b = 11 \), \( c = 12 \). \( a \times c = 2 \times 12 = 24 \). We need two numbers that multiply to 24 and add to 11. Possible pairs: (1, 24) sum 25, (2, 12) sum 14, (3, 8) sum 11. Yes, 3 and 8. --- **Step 4: Rewrite the middle term** \( 2x^2 + 11x + 12 = 2x^2 + 3x + 8x + 12 \). --- **Step 5: Factor by grouping** Group: \( (2x^2 + 3x) + (8x + 12) \). Factor each group: From \( 2x^2 + 3x \), factor \( x \): \( x(2x + 3) \). From \( 8x + 12 \), factor \( 4 \): \( 4(2x + 3) \). So we have: \( x(2x + 3) + 4(2x + 3) \). --- **Step 6: Factor out the common binomial** \( (2x + 3)(x + 4) \). So the factored form is \( (2x + 3)(x + 4) \). --- **Step 7: Interpret dimensions** Area = length × width = \( (2x + 3)(x + 4) \). We know length is 5 more than width. Let width = \( x + 4 \) and length = \( 2x + 3 \). Check if length = width + 5: \( 2x + 3 = (x + 4) + 5 \) \( 2x + 3 = x + 9 \) \( x = 6 \). --- **Step 8: Find actual dimensions** If \( x = 6 \): Width = \( x + 4 = 10 \) m. Length = \( 2x + 3 = 15 \) m. Check: 15 = 10 + 5, correct. Area = \( 15 \times 10 = 150 \) m². Check original expression: \( 2(6)^2 + 11(6) + 12 = 72 + 66 + 12 = 150 \), correct. --- **Step 9: Final answer** The factored form representing length and width is \( (2x + 3)(x + 4) \). So the correct answer is: (2x + 3)(x + 4)

  2. Factor: 14x² + 39x + 10 Answer: (7x + 2)(2x + 5) Solution: Identify a = 14, b = 39, c = 10 Multiply a and c: 14 × 10 = 140 Find two numbers that multiply to 140 and add to 39: 35 and 4 (since 35 × 4 = 140 and 35 + 4 = 39) Rewrite the middle term: 14x² + 35x + 4x + 10 Factor by grouping: (14x² + 35x) + (4x + 10) Factor out common factors: 7x(2x + 5) +…
    Full step-by-step solution

    Step 1: Identify a = 14, b = 39, c = 10 Step 2: Multiply a and c: 14 × 10 = 140 Step 3: Find two numbers that multiply to 140 and add to 39: 35 and 4 (since 35 × 4 = 140 and 35 + 4 = 39) Step 4: Rewrite the middle term: 14x² + 35x + 4x + 10 Step 5: Factor by grouping: (14x² + 35x) + (4x + 10) Step 6: Factor out common factors: 7x(2x + 5) + 2(2x + 5) Step 7: Factor out the common binomial: (2x + 5)(7x + 2) The answer is (7x + 2)(2x + 5).

  3. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the equation h(t) = -5t² + 18t + 8, where h is height in meters and t is time in seconds. To analyze when the ball hits the ground, they need to factor this quadratic expression using the AC method. What is the factored form of -5t² + 18t + 8? Answer: (-5t - 2)(t - 4) Solution: The AC method involves finding two numbers that multiply to the product of the first and last coefficients and add to the middle coefficient. For quadratics with negative leading coefficients, you can either factor out -1 first or work carefully with negative numbers.
    Full step-by-step solution

    The AC method involves finding two numbers that multiply to the product of the first and last coefficients and add to the middle coefficient. For quadratics with negative leading coefficients, you can either factor out -1 first or work carefully with negative numbers. This method is particularly useful for analyzing projectile motion equations in physics, where the factored form helps identify key points like when an object reaches certain heights or returns to the ground.

  4. A physics class is designing a catapult that launches projectiles. The height of a projectile above ground is modeled by the quadratic equation h(t) = -16t² + 48t + 64, where h is height in feet and t is time in seconds. To analyze the projectile's flight time, factor this quadratic expression using the AC method to determine when the projectile will return to ground level. Answer: (t - 4)(-16t - 16) Solution: The AC method for factoring quadratics involves finding two numbers that multiply to the product of the first and last coefficients (A × C) and add to the middle coefficient (B). For quadratics with negative leading coefficients, it's often helpful to factor out -1 first.
    Full step-by-step solution

    The AC method for factoring quadratics involves finding two numbers that multiply to the product of the first and last coefficients (A × C) and add to the middle coefficient (B). For quadratics with negative leading coefficients, it's often helpful to factor out -1 first. This method is particularly useful in physics applications like projectile motion, where we need to find key points in time such as when an object reaches maximum height or returns to its starting position.

  5. Factor: 8x² + 26x + 15 = ? Answer: (4x + 3)(2x + 5) 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 common factors: 4x(2x + 5) + 3(2x + 5). Step 7: Factor out the common binomial: (2x + 5)(4x + 3). The answer is (4x + 3)(2x + 5).

  6. 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.

  7. Factor: 6x² + 31x + 35 Answer: (3x + 5)(2x + 7) Solution: Identify a = 6, b = 31, c = 35. Multiply a and c: 6 × 35 = 210. Find two numbers that multiply to 210 and add to 31.
    Full step-by-step solution

    Step 1: Identify a = 6, b = 31, c = 35. Step 2: Multiply a and c: 6 × 35 = 210. Step 3: Find two numbers that multiply to 210 and add to 31. The numbers are 21 and 10 because 21 × 10 = 210 and 21 + 10 = 31. Step 4: Rewrite the middle term: 6x² + 21x + 10x + 35. Step 5: Factor by grouping: (6x² + 21x) + (10x + 35). Step 6: Factor out common factors: 3x(2x + 7) + 5(2x + 7). Step 7: Factor out the common binomial: (2x + 7)(3x + 5). The answer is (3x + 5)(2x + 7).

  8. Factor: 4x² + 12x + 9 = ? Answer: (2x + 3)² Solution: A perfect square trinomial follows the pattern a² + 2ab + b² = (a + b)². Check if the middle term equals 2 times the product of the square roots of the first and last terms.