Factoring Quadratic Expressions

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Factoring Quadratic Expressions · Grade 10 · Worksheet 3

1. Emma is designing a rectangular flower bed in her backyard. The area of the flower bed can be modeled by the quadratic expression 3x² + 23x + 14 square feet, where x is a positive integer representing the width in feet. If the length of the flower bed is (3x + 2) feet, what expression represents the width of the flower bed in terms of x? Answer: (x + 7) feet Solution: The area of a rectangle is length times width. We are given area = 3x² + 23x + 14 and length = 3x + 2. To find the width, we need to divide the area by the length: width = (3x² + 23x + 14) / (3x + 2).
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. We are given area = 3x² + 23x + 14 and length = 3x + 2. Step 2: To find the width, we need to divide the area by the length: width = (3x² + 23x + 14) / (3x + 2). Step 3: Factor the quadratic expression 3x² + 23x + 14. We need two numbers that multiply to (3)(14) = 42 and add to 23. The numbers are 21 and 2. Step 4: Rewrite the middle term: 3x² + 21x + 2x + 14. Step 5: Factor by grouping: (3x² + 21x) + (2x + 14) = 3x(x + 7) + 2(x + 7) = (3x + 2)(x + 7). Step 6: So area = (3x + 2)(x + 7). Since length = 3x + 2, the width must be x + 7. Step 7: Therefore, the width is (x + 7) feet. The answer is (x + 7) feet.

2. A right triangle is positioned on a coordinate plane with vertices at (0,0), (2x,0), and (0,x-3). The area of this triangle is 20 square units. Find the positive value of x that satisfies this condition. Answer: 8 Solution: The triangle has vertices at (0,0), (2x,0), and (0,x-3). Since the legs are along the axes, the base is 2x and the height is (x-3). The area of a triangle is (1/2) × base × height.
   Full step-by-step solution

   Step 1: The triangle has vertices at (0,0), (2x,0), and (0,x-3). Since the legs are along the axes, the base is 2x and the height is (x-3). Step 2: The area of a triangle is (1/2) × base × height. Step 3: Set up the equation: (1/2) × (2x) × (x-3) = 20 Step 4: Simplify: (1/2) × 2x × (x-3) = x(x-3) = 20 Step 5: Expand: x² - 3x = 20 Step 6: Rearrange to standard quadratic form: x² - 3x - 20 = 0 Step 7: Factor the quadratic: (x-8)(x+5) = 0 Step 8: Solve for x: x = 8 or x = -5 Step 9: Since x represents a length and must be positive, we take x = 8. The positive value of x is 8.

3. Factor: 12x² - 17x - 7 = ? Answer: (4x - 7)(3x + 1) Solution: Identify a = 12, b = -17, c = -7. Multiply a and c: 12 × (-7) = -84. Find two numbers that multiply to -84 and add to -17: -21 and 4 (since -21 × 4 = -84 and -21 + 4 = -17).
   Full step-by-step solution

   Step 1: Identify a = 12, b = -17, c = -7. Step 2: Multiply a and c: 12 × (-7) = -84. Step 3: Find two numbers that multiply to -84 and add to -17: -21 and 4 (since -21 × 4 = -84 and -21 + 4 = -17). Step 4: Rewrite the middle term: 12x² - 21x + 4x - 7. Step 5: Group terms: (12x² - 21x) + (4x - 7). Step 6: Factor each group: 3x(4x - 7) + 1(4x - 7). Step 7: Factor out the common binomial: (4x - 7)(3x + 1). The factored form is (4x - 7)(3x + 1).

4. Factor: 15x² + 35x + 20 Answer: 5(3x + 4)(x + 1) Solution: Factor out the greatest common factor (GCF) of 5: 15x² + 35x + 20 = 5(3x² + 7x + 4). Factor the trinomial 3x² + 7x + 4. Multiply the leading coefficient (3) by the constant term (4): 3 × 4 = 12.
   Full step-by-step solution

   Step 1: Factor out the greatest common factor (GCF) of 5: 15x² + 35x + 20 = 5(3x² + 7x + 4). Step 2: Factor the trinomial 3x² + 7x + 4. Multiply the leading coefficient (3) by the constant term (4): 3 × 4 = 12. Step 3: Find two numbers that multiply to 12 and add to 7: 3 and 4. Step 4: Rewrite the middle term: 3x² + 3x + 4x + 4. Step 5: Factor by grouping: (3x² + 3x) + (4x + 4) = 3x(x + 1) + 4(x + 1). Step 6: Factor out the common binomial (x + 1): (3x + 4)(x + 1). Step 7: Include the GCF: 5(3x + 4)(x + 1). The factored form is 5(3x + 4)(x + 1).

5. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 28x - 80, where P(x) is the profit in thousands of dollars. The company breaks even when profit is zero. At what production levels does the company break even? Answer: 4 and 10 Solution: Set the profit function equal to zero since break-even occurs when profit is zero: -2x² + 28x - 80 = 0 Multiply both sides by -1 to make the leading coefficient positive: 2x² - 28x + 80 = 0 Divide all terms by 2 to simplify: x² - 14x + 40 = 0 Factor the quadratic: (x - 4)(x - 10) = 0 Set each…
   Full step-by-step solution

   Step 1: Set the profit function equal to zero since break-even occurs when profit is zero: -2x² + 28x - 80 = 0 Step 2: Multiply both sides by -1 to make the leading coefficient positive: 2x² - 28x + 80 = 0 Step 3: Divide all terms by 2 to simplify: x² - 14x + 40 = 0 Step 4: Factor the quadratic: (x - 4)(x - 10) = 0 Step 5: Set each factor equal to zero: x - 4 = 0 or x - 10 = 0 Step 6: Solve for x: x = 4 or x = 10 The company breaks even at production levels of 4 units and 10 units.

6. A physics student is analyzing the trajectory of a projectile launched from ground level. The height of the projectile (in meters) at time t seconds is modeled by the quadratic function h(t) = -5t² + 20t. The student wants to determine the time interval during which the projectile is at least 15 meters above the ground. Which inequality represents this situation? Answer: C. -5t² + 20t ≥ 15 Solution: The height function is h(t) = -5t² + 20t 'At least 15 meters' means the height must be greater than or equal to 15 This gives us the inequality: -5t² + 20t ≥ 15 To solve this, we would rearrange to: -5t² + 20t - 15 ≥ 0 Multiply by -1 (reverse inequality): 5t² - 20t + 15 ≤ 0 Divide by 5: t² - 4t…
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 20t Step 2: 'At least 15 meters' means the height must be greater than or equal to 15 Step 3: This gives us the inequality: -5t² + 20t ≥ 15 Step 4: To solve this, we would rearrange to: -5t² + 20t - 15 ≥ 0 Step 5: Multiply by -1 (reverse inequality): 5t² - 20t + 15 ≤ 0 Step 6: Divide by 5: t² - 4t + 3 ≤ 0 Step 7: Factor: (t - 1)(t - 3) ≤ 0 Step 8: The solution is 1 ≤ t ≤ 3 seconds Step 9: The correct inequality that represents the situation is -5t² + 20t ≥ 15

7. Factor: 12x² + 17x + 6 Answer: (3x + 2)(4x + 3) Solution: Identify a = 12, b = 17, c = 6. Multiply a and c: 12 × 6 = 72. Find two numbers that multiply to 72 and add to 17.
   Full step-by-step solution

   Step 1: Identify a = 12, b = 17, c = 6. Step 2: Multiply a and c: 12 × 6 = 72. Step 3: Find two numbers that multiply to 72 and add to 17. These numbers are 8 and 9 because 8 × 9 = 72 and 8 + 9 = 17. Step 4: Rewrite the middle term using these numbers: 12x² + 8x + 9x + 6. Step 5: Factor by grouping: (12x² + 8x) + (9x + 6). Step 6: Factor out the greatest common factor from each group: 4x(3x + 2) + 3(3x + 2). Step 7: Factor out the common binomial (3x + 2): (3x + 2)(4x + 3). The factored form is (3x + 2)(4x + 3).

8. Factor completely: 6x² - 13x - 5 = 0 Answer: (2x - 5)(3x + 1) Solution: Multiply the leading coefficient (6) by the constant term (-5): 6 × (-5) = -30 Find two numbers that multiply to -30 and add to -13: -15 and 2 Rewrite the middle term: 6x² - 15x + 2x - 5 = 0 Factor by grouping: (6x² - 15x) + (2x - 5) = 0 Factor out common factors: 3x(2x - 5) + 1(2x - 5) = 0…
   Full step-by-step solution

   Step 1: Multiply the leading coefficient (6) by the constant term (-5): 6 × (-5) = -30 Step 2: Find two numbers that multiply to -30 and add to -13: -15 and 2 Step 3: Rewrite the middle term: 6x² - 15x + 2x - 5 = 0 Step 4: Factor by grouping: (6x² - 15x) + (2x - 5) = 0 Step 5: Factor out common factors: 3x(2x - 5) + 1(2x - 5) = 0 Step 6: Factor out the common binomial: (2x - 5)(3x + 1) = 0 The completely factored form is (2x - 5)(3x + 1).