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Factoring Quadratic Expressions

Grade 10 · Mathematics · Worksheet 3

  1. Aroha is designing a rectangular kapa (traditional cloth) for a cultural exhibition. The area of the kapa in square meters is given by the quadratic expression 9x² + 39x + 42, where x is a positive integer representing a scaling factor. The length of the kapa is known to be (3x + 7) meters. What expression represents the width of the kapa in meters? Answer: ______________
  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of the triangle is 16 square units. Find the value of x. Answer: ______________
  3. Factor completely: 6x² - 13x - 5 = 0 Answer: ______________
  4. A company's profit from selling a new smartphone can be modeled by the quadratic function P(x) = -2x² + 120x - 1600, where x represents the number of phones sold (in hundreds) and P(x) is the profit in thousands of dollars. The company needs to determine the break-even points where profit equals zero. At what production levels does the company break even? Answer: ______________
  5. Factor: 12x² + 17x + 6 Answer: ______________
  6. 3x² - 14x - 5 = 0 Answer: ______________
  7. Factor: 15x² + 50x + 35 Answer: ______________
  8. Aroha is designing a rectangular garden for a community project. The area of the garden (in square meters) is given by the quadratic expression 3x² + 17x + 20, where x is a positive integer representing the width in meters. If the length of the garden is (3x + 5) meters, what expression represents the width of the garden in terms of x? Answer: ______________
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Answer Key & Explanations

Factoring Quadratic Expressions · Grade 10 · Worksheet 3

  1. Aroha is designing a rectangular kapa (traditional cloth) for a cultural exhibition. The area of the kapa in square meters is given by the quadratic expression 9x² + 39x + 42, where x is a positive integer representing a scaling factor. The length of the kapa is known to be (3x + 7) meters. What expression represents the width of the kapa in meters? Answer: 3x + 6 Solution: The area of a rectangle is length times width. We are given area = 9x² + 39x + 42 and length = (3x + 7). To find the width, we need to factor the quadratic expression 9x² + 39x + 42 so that one factor is (3x + 7).
    Full step-by-step solution

    Step 1: The area of a rectangle is length times width. We are given area = 9x² + 39x + 42 and length = (3x + 7). To find the width, we need to factor the quadratic expression 9x² + 39x + 42 so that one factor is (3x + 7). Step 2: Factor 9x² + 39x + 42. First, notice that each term has a common factor of 3: 9x² + 39x + 42 = 3(3x² + 13x + 14). Step 3: Now factor the quadratic inside: 3x² + 13x + 14. Multiply the coefficient of x² (3) by the constant term (14): 3 × 14 = 42. We need two numbers that multiply to 42 and add to 13 (the coefficient of x). The numbers are 6 and 7 because 6 × 7 = 42 and 6 + 7 = 13. Step 4: Rewrite the middle term using these numbers: 3x² + 6x + 7x + 14. Step 5: Group the terms: (3x² + 6x) + (7x + 14). Step 6: Factor each group: 3x(x + 2) + 7(x + 2). Step 7: Factor out the common binomial (x + 2): (x + 2)(3x + 7). Step 8: So the original expression becomes: 3(x + 2)(3x + 7) = (3x + 6)(3x + 7). Step 9: Since the area is (3x + 6)(3x + 7) and the length is (3x + 7), the width must be (3x + 6). The answer is 3x + 6.

  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of the triangle is 16 square units. Find the value of x. Answer: 4 Solution: The vertices are at (0,0), (x,0), and (0,2x). This means one leg of the right triangle is along the x-axis from (0,0) to (x,0), so its length is x.
    Full step-by-step solution

    Step 1: Understand the triangle's vertices. The vertices are at (0,0), (x,0), and (0,2x). This means one leg of the right triangle is along the x-axis from (0,0) to (x,0), so its length is x. The other leg is along the y-axis from (0,0) to (0,2x), so its length is 2x. Step 2: Identify the base and height. Since it's a right triangle with legs along the axes, we can take the leg along the x-axis as the base and the leg along the y-axis as the height. Base = x Height = 2x Step 3: Write the formula for the area of a triangle. Area = (1/2) * base * height Step 4: Substitute the known area and the expressions for base and height. Area = 16 So: 16 = (1/2) * (x) * (2x) Step 5: Simplify the equation. 16 = (1/2) * 2x * x 16 = (1/2) * 2x^2 16 = x^2 Step 6: Solve for x. x^2 = 16 x = 4 or x = -4 Step 7: Interpret the solution. Since x is a length (distance along the x-axis), it must be positive. Therefore, x = 4. Final answer: x = 4

  3. 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).

  4. A company's profit from selling a new smartphone can be modeled by the quadratic function P(x) = -2x² + 120x - 1600, where x represents the number of phones sold (in hundreds) and P(x) is the profit in thousands of dollars. The company needs to determine the break-even points where profit equals zero. At what production levels does the company break even? Answer: x = 20 or x = 40 Solution: Break-even analysis in business involves finding when revenue equals costs, resulting in zero profit. For quadratic profit functions, this means solving a quadratic equation.
    Full step-by-step solution

    Break-even analysis in business involves finding when revenue equals costs, resulting in zero profit. For quadratic profit functions, this means solving a quadratic equation. The solutions represent the production levels where the company neither makes nor loses money. Factoring quadratics is a common algebraic technique used to find these critical business points.

  5. 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).

  6. 3x² - 14x - 5 = 0 Answer: x = 5, x = -1/3 Solution: Multiply the leading coefficient (3) by the constant term (-5): 3 × (-5) = -15 Find two numbers that multiply to -15 and add to -14: -15 and 1 Rewrite the middle term: 3x² - 15x + x - 5 = 0 Factor by grouping: (3x² - 15x) + (x - 5) = 0 Factor out common factors: 3x(x - 5) + 1(x - 5) = 0 Factor…
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (3) by the constant term (-5): 3 × (-5) = -15 Step 2: Find two numbers that multiply to -15 and add to -14: -15 and 1 Step 3: Rewrite the middle term: 3x² - 15x + x - 5 = 0 Step 4: Factor by grouping: (3x² - 15x) + (x - 5) = 0 Step 5: Factor out common factors: 3x(x - 5) + 1(x - 5) = 0 Step 6: Factor out the common binomial: (3x + 1)(x - 5) = 0 Step 7: Set each factor equal to zero: 3x + 1 = 0 or x - 5 = 0 Step 8: Solve each equation: x = -1/3 or x = 5 Step 9: The solutions are x = -1/3 and x = 5

  7. Factor: 15x² + 50x + 35 Answer: 5(3x + 7)(x + 1) Solution: Identify the GCF of 15x², 50x, and 35. The GCF is 5. Factor out the GCF: 5(3x² + 10x + 7).
    Full step-by-step solution

    Step 1: Identify the GCF of 15x², 50x, and 35. The GCF is 5. Step 2: Factor out the GCF: 5(3x² + 10x + 7). Step 3: Now factor the trinomial 3x² + 10x + 7. Multiply a and c: 3 × 7 = 21. Step 4: Find two numbers that multiply to 21 and add to 10: 7 and 3 (since 7 × 3 = 21 and 7 + 3 = 10). Step 5: Rewrite the middle term: 3x² + 7x + 3x + 7. Step 6: Factor by grouping: (3x² + 7x) + (3x + 7). Step 7: Factor out the GCF from each group: x(3x + 7) + 1(3x + 7). Step 8: Factor out the common binomial (3x + 7): (3x + 7)(x + 1). Step 9: Include the GCF: 5(3x + 7)(x + 1). The factored form is 5(3x + 7)(x + 1).

  8. Aroha is designing a rectangular garden for a community project. The area of the garden (in square meters) is given by the quadratic expression 3x² + 17x + 20, where x is a positive integer representing the width in meters. If the length of the garden is (3x + 5) meters, what expression represents the width of the garden in terms of x? Answer: (x + 4) Solution: The area of a rectangle is length times width. We are given area = 3x² + 17x + 20 and length = (3x + 5). To find the width, we need to divide the area by the length: width = area / length = (3x² + 17x + 20) / (3x + 5).
    Full step-by-step solution

    Step 1: The area of a rectangle is length times width. We are given area = 3x² + 17x + 20 and length = (3x + 5). Step 2: To find the width, we need to divide the area by the length: width = area / length = (3x² + 17x + 20) / (3x + 5). Step 3: Factor the quadratic expression 3x² + 17x + 20. We look for two numbers that multiply to 3 * 20 = 60 and add to 17. The numbers are 12 and 5. Step 4: Rewrite the middle term: 3x² + 12x + 5x + 20. Step 5: Group terms: (3x² + 12x) + (5x + 20). Step 6: Factor out the GCF from each group: 3x(x + 4) + 5(x + 4). Step 7: Factor out the common binomial (x + 4): (x + 4)(3x + 5). Step 8: So, 3x² + 17x + 20 = (x + 4)(3x + 5). Step 9: Now, width = (3x² + 17x + 20) / (3x + 5) = (x + 4)(3x + 5) / (3x + 5) = x + 4. The answer is (x + 4).