Factor Trinomials

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Factor Trinomials · Grade 9 · Worksheet 2

1. 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? Use the AC method to factor the quadratic expression. Answer: (x + 2) Solution: Area = 2x² + 7x + 6 Length = (2x + 3) We want Width. Write the relationship. Area = Length × Width So Width = Area ÷ Length = (2x² + 7x + 6) ÷ (2x + 3) Factor the quadratic using the AC method.
   Full step-by-step solution

   We are given: Area = 2x² + 7x + 6 Length = (2x + 3) We want Width. Step 1: Write the relationship. Area = Length × Width So Width = Area ÷ Length = (2x² + 7x + 6) ÷ (2x + 3) Step 2: Factor the quadratic using the AC method. Quadratic: 2x² + 7x + 6 Multiply A × C = 2 × 6 = 12 We need two numbers that multiply to 12 and add to 7 (the coefficient of x). Those numbers are 3 and 4 because 3 × 4 = 12 and 3 + 4 = 7. Step 3: Rewrite the middle term using these numbers. 2x² + 3x + 4x + 6 Step 4: Factor by grouping. Group terms: (2x² + 3x) + (4x + 6) Factor each group: x(2x + 3) + 2(2x + 3) Step 5: Factor out the common binomial. (2x + 3)(x + 2) Step 6: Write the width. Width = Area ÷ Length = [(2x + 3)(x + 2)] ÷ (2x + 3) Cancel (2x + 3) (assuming x ≠ -3/2 so length ≠ 0): Width = x + 2 Final answer: x + 2

2. Factor: 15x² + 31x + 10 Answer: (5x + 2)(3x + 5) Solution: Identify a = 15, b = 31, c = 10 Multiply a and c: 15 × 10 = 150 Find two numbers that multiply to 150 and add to 31: 25 and 6 (since 25 × 6 = 150 and 25 + 6 = 31) Rewrite the middle term: 15x² + 25x + 6x + 10 Factor by grouping: (15x² + 25x) + (6x + 10) Factor out common factors: 5x(3x + 5) +…
   Full step-by-step solution

   Step 1: Identify a = 15, b = 31, c = 10 Step 2: Multiply a and c: 15 × 10 = 150 Step 3: Find two numbers that multiply to 150 and add to 31: 25 and 6 (since 25 × 6 = 150 and 25 + 6 = 31) Step 4: Rewrite the middle term: 15x² + 25x + 6x + 10 Step 5: Factor by grouping: (15x² + 25x) + (6x + 10) Step 6: Factor out common factors: 5x(3x + 5) + 2(3x + 5) Step 7: Factor out the common binomial: (3x + 5)(5x + 2) The answer is (5x + 2)(3x + 5).

3. Factor: 2x² + 7x + 3 = ? Answer: (2x + 1)(x + 3) Solution: Identify coefficients. The expression is in the form ax² + bx + c, where: a = 2 b = 7 c = 3 Multiply a and c. a * c = 2 * 3 = 6 Find two numbers that multiply to 6 and add to b (which is 7).
   Full step-by-step solution

   Let's factor the quadratic expression: 2x² + 7x + 3. Step 1: Identify coefficients. The expression is in the form ax² + bx + c, where: a = 2 b = 7 c = 3 Step 2: Multiply a and c. a * c = 2 * 3 = 6 Step 3: Find two numbers that multiply to 6 and add to b (which is 7). List factor pairs of 6: 1 and 6 → 1 + 6 = 7 ← This works. 2 and 3 → 2 + 3 = 5 ← Not 7. So the numbers are 1 and 6. Step 4: Rewrite the middle term (7x) using the two numbers found. 2x² + 7x + 3 = 2x² + 1x + 6x + 3 Step 5: Factor by grouping. Group the first two terms and the last two terms: (2x² + 1x) + (6x + 3) Factor out the greatest common factor from each group: From (2x² + 1x), factor out x: x(2x + 1) From (6x + 3), factor out 3: 3(2x + 1) Now we have: x(2x + 1) + 3(2x + 1) Step 6: Factor out the common binomial (2x + 1). (2x + 1)(x + 3) Step 7: Check by multiplying (FOIL). First: 2x * x = 2x² Outer: 2x * 3 = 6x Inner: 1 * x = x Last: 1 * 3 = 3 Combine: 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓ Final answer: (2x + 1)(x + 3)

4. A rectangular skatepark has an area represented by the quadratic expression 3x² + 14x + 8 square meters, where x represents the width in meters. The length of the skatepark is 2 meters more than three times its width. Factor the quadratic expression using the AC method to determine the binomial expressions that represent the skatepark's dimensions. Answer: (3x+2)(x+4) Solution: The AC method is a systematic approach to factoring quadratic trinomials of the form ax² + bx + c. It involves finding two numbers that multiply to the product of a and c, and add to b. This technique is particularly useful when the leading coefficient is not 1, as it provides a reliable method…
   Full step-by-step solution

   The AC method is a systematic approach to factoring quadratic trinomials of the form ax² + bx + c. It involves finding two numbers that multiply to the product of a and c, and add to b. These numbers are used to split the middle term, allowing for factoring by grouping. This technique is particularly useful when the leading coefficient is not 1, as it provides a reliable method for determining the correct binomial factors.

5. Factor: 8x² + 14x - 15 Answer: (4x - 3)(2x + 5) Solution: Identify a = 8, b = 14, c = -15. Multiply a and c: 8 × (-15) = -120. Find two numbers that multiply to -120 and add to 14.
   Full step-by-step solution

   Step 1: Identify a = 8, b = 14, c = -15. Step 2: Multiply a and c: 8 × (-15) = -120. Step 3: Find two numbers that multiply to -120 and add to 14. The numbers are 20 and -6 because 20 × (-6) = -120 and 20 + (-6) = 14. Step 4: Rewrite the middle term using these numbers: 8x² + 20x - 6x - 15. Step 5: Factor by grouping: (8x² + 20x) + (-6x - 15). Step 6: Factor out the greatest common factor from each group: 4x(2x + 5) - 3(2x + 5). Step 7: Factor out the common binomial (2x + 5): (2x + 5)(4x - 3). The answer is (4x - 3)(2x + 5).

6. Factor: 21x² + 47x + 20 Answer: (7x + 5)(3x + 4) Solution: Identify a = 21, b = 47, c = 20. Multiply a and c: 21 × 20 = 420. Find two numbers that multiply to 420 and add to 47.
   Full step-by-step solution

   Step 1: Identify a = 21, b = 47, c = 20. Step 2: Multiply a and c: 21 × 20 = 420. Step 3: Find two numbers that multiply to 420 and add to 47. The numbers are 35 and 12 because 35 × 12 = 420 and 35 + 12 = 47. Step 4: Rewrite the middle term: 21x² + 35x + 12x + 20. Step 5: Factor by grouping: (21x² + 35x) + (12x + 20). Step 6: Factor out common factors: 7x(3x + 5) + 4(3x + 5). Step 7: Factor out the common binomial: (3x + 5)(7x + 4). The answer is (7x + 5)(3x + 4).

7. Factor: 10x² + 35x + 30 = ? Answer: 5(2x + 3)(x + 2) Solution: Identify a = 10, b = 35, c = 30. Check for a common factor: all coefficients are divisible by 5. Factor out 5: 5(2x² + 7x + 6).
   Full step-by-step solution

   Step 1: Identify a = 10, b = 35, c = 30. Check for a common factor: all coefficients are divisible by 5. Factor out 5: 5(2x² + 7x + 6). Step 2: Now factor the quadratic 2x² + 7x + 6 using the AC method. Here a = 2, b = 7, c = 6. Step 3: Multiply a and c: 2 × 6 = 12. Step 4: Find two numbers that multiply to 12 and add to 7: 3 and 4 (since 3 × 4 = 12 and 3 + 4 = 7). Step 5: Rewrite the middle term: 2x² + 3x + 4x + 6. Step 6: Factor by grouping: (2x² + 3x) + (4x + 6) = x(2x + 3) + 2(2x + 3). Step 7: Factor out the common binomial: (2x + 3)(x + 2). Step 8: Include the common factor from Step 1: 5(2x + 3)(x + 2). The answer is 5(2x + 3)(x + 2).

8. Factor: 12x² - 7x - 10 = ? Answer: (4x - 5)(3x + 2) Solution: Identify a = 12, b = -7, c = -10 Multiply a × c = 12 × (-10) = -120 Find two numbers that multiply to -120 and add to -7: -15 and 8 Rewrite the middle term: 12x² - 15x + 8x - 10 Factor by grouping: (12x² - 15x) + (8x - 10) Factor out common factors: 3x(4x - 5) + 2(4x - 5) Factor out the common…
   Full step-by-step solution

   Step 1: Identify a = 12, b = -7, c = -10 Step 2: Multiply a × c = 12 × (-10) = -120 Step 3: Find two numbers that multiply to -120 and add to -7: -15 and 8 Step 4: Rewrite the middle term: 12x² - 15x + 8x - 10 Step 5: Factor by grouping: (12x² - 15x) + (8x - 10) Step 6: Factor out common factors: 3x(4x - 5) + 2(4x - 5) Step 7: Factor out the common binomial: (4x - 5)(3x + 2) The answer is (4x - 5)(3x + 2).

9. A rectangular garden has an area of 6x² + 19x + 10 square feet. If the length of the garden is represented by (2x + 5) feet, what is the width of the garden in factored form? Answer: (3x + 2) Solution: Area = 6x² + 19x + 10 square feet Length = (2x + 5) feet Width = ? Recall the formula for the area of a rectangle. Area = Length × Width So Width = Area ÷ Length Write the division expression.
   Full step-by-step solution

   We are given: Area = 6x² + 19x + 10 square feet Length = (2x + 5) feet Width = ? Step 1: Recall the formula for the area of a rectangle. Area = Length × Width So Width = Area ÷ Length Step 2: Write the division expression. Width = (6x² + 19x + 10) ÷ (2x + 5) Step 3: Factor the quadratic expression 6x² + 19x + 10. We look for two numbers that multiply to 6×10 = 60 and add to 19. The numbers are 15 and 4 because 15×4 = 60 and 15+4 = 19. Step 4: Rewrite the middle term using 15 and 4. 6x² + 15x + 4x + 10 Step 5: Factor by grouping. Group the first two terms: 6x² + 15x = 3x(2x + 5) Group the last two terms: 4x + 10 = 2(2x + 5) So we have: 3x(2x + 5) + 2(2x + 5) Step 6: Factor out the common factor (2x + 5). (2x + 5)(3x + 2) Step 7: Now the division becomes: Width = (2x + 5)(3x + 2) ÷ (2x + 5) Step 8: Cancel the common factor (2x + 5) (assuming x ≠ -5/2 so length ≠ 0). Width = 3x + 2 Final answer: (3x + 2)