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Factor Quadratics

Grade 9 · Algebra · Worksheet 1

  1. A local theater is designing a new stage backdrop that will have a rectangular opening for special effects. The area of the opening is given by the quadratic expression 3x² + 10x + 8 square feet, where x represents the additional length beyond the minimum dimensions. The theater director knows that the width of the opening is (x + 2) feet. What expression represents the length of the opening in factored form? Answer: ______________
  2. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden is 65 square meters. If you were to draw this garden on a coordinate plane with one corner at the origin and the sides along the axes, what would be the dimensions of the rectangle? Answer: ______________
  3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,x-3). The area of the triangle is 14 square units. Write the quadratic equation in standard form that represents the relationship between x and the given conditions. Answer: ______________
  4. Aroha is designing a rectangular pathway for a community garden. The area of the pathway is given by the quadratic expression 15x² + 34x + 15 square meters, where x represents the number of extra meters beyond the minimum width. The length of the pathway is (3x + 5) meters. What expression in factored form represents the width of the pathway? Answer: ______________
  5. x² - 9x + 20 = ? Answer: ______________
  6. Factor completely: 7x² - 27x + 18 = ? Answer: ______________
  7. Factor completely: 12x² + 37x + 21 = ? Answer: ______________
  8. Factor completely: 14x² - 41x + 15 = ? Answer: ______________
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Answer Key & Explanations

Factor Quadratics · Grade 9 · Worksheet 1

  1. A local theater is designing a new stage backdrop that will have a rectangular opening for special effects. The area of the opening is given by the quadratic expression 3x² + 10x + 8 square feet, where x represents the additional length beyond the minimum dimensions. The theater director knows that the width of the opening is (x + 2) feet. What expression represents the length of the opening in factored form? Answer: (3x + 4) Solution: In geometry, the area of a rectangle equals length times width. When working with polynomial expressions for area, if one dimension is known, the other can be found by factoring or dividing the area expression.
    Full step-by-step solution

    In geometry, the area of a rectangle equals length times width. When working with polynomial expressions for area, if one dimension is known, the other can be found by factoring or dividing the area expression. This concept applies to various real-world situations like calculating dimensions of rooms, gardens, or architectural elements when the total area and one measurement are known.

  2. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden is 65 square meters. If you were to draw this garden on a coordinate plane with one corner at the origin and the sides along the axes, what would be the dimensions of the rectangle? Answer: width = 5 meters, length = 13 meters Solution: Let the width of the rectangle be \( w \) meters. The length is 3 meters more than twice the width, so length \( l = 2w + 3 \).
    Full step-by-step solution

    Let's go step by step. --- **Step 1: Define variables** Let the width of the rectangle be \( w \) meters. The length is 3 meters more than twice the width, so length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area = length × width \( 65 = l \times w \) Substitute \( l = 2w + 3 \): \( 65 = (2w + 3) \times w \) --- **Step 3: Expand and rearrange** \( 65 = 2w^2 + 3w \) Bring all terms to one side: \( 2w^2 + 3w - 65 = 0 \) --- **Step 4: Solve the quadratic equation** We can use factoring: We need two numbers whose product is \( 2 \times (-65) = -130 \) and whose sum is \( 3 \). The numbers are \( 13 \) and \( -10 \), because \( 13 \times (-10) = -130 \) and \( 13 + (-10) = 3 \). Rewrite \( 3w \) as \( 13w - 10w \): \( 2w^2 + 13w - 10w - 65 = 0 \) Group terms: \( w(2w + 13) - 5(2w + 13) = 0 \) Factor: \( (2w + 13)(w - 5) = 0 \) --- **Step 5: Find possible values of w** \( 2w + 13 = 0 \) → \( w = -6.5 \) (not valid, width can't be negative) \( w - 5 = 0 \) → \( w = 5 \) (valid) So width \( w = 5 \) meters. --- **Step 6: Find length** \( l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \) meters. --- **Step 7: Check area** Area = \( 13 \times 5 = 65 \) square meters. Correct. --- **Final answer:** width = 5 meters, length = 13 meters.

  3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,x-3). The area of the triangle is 14 square units. Write the quadratic equation in standard form that represents the relationship between x and the given conditions. Answer: x^2 - 3x - 28 = 0 Solution: - The base is along the x-axis from (0,0) to (x,0), so base = x - The height is along the y-axis from (0,0) to (0,x-3), so height = x-3 Area = (1/2) × base × height 14 = (1/2) × x × (x-3) 14 = (1/2)(x)(x-3) Multiply both sides by 2: 28 = x(x-3) 28 = x^2 - 3x x^2 - 3x - 28 = 0 The quadratic…
    Full step-by-step solution

    Step 1: Identify the base and height from the coordinates - The base is along the x-axis from (0,0) to (x,0), so base = x - The height is along the y-axis from (0,0) to (0,x-3), so height = x-3 Step 2: Apply the area formula for a triangle Area = (1/2) × base × height 14 = (1/2) × x × (x-3) Step 3: Simplify the equation 14 = (1/2)(x)(x-3) Multiply both sides by 2: 28 = x(x-3) Step 4: Expand and rearrange into standard quadratic form 28 = x^2 - 3x x^2 - 3x - 28 = 0 The quadratic equation in standard form is x^2 - 3x - 28 = 0.

  4. Aroha is designing a rectangular pathway for a community garden. The area of the pathway is given by the quadratic expression 15x² + 34x + 15 square meters, where x represents the number of extra meters beyond the minimum width. The length of the pathway is (3x + 5) meters. What expression in factored form represents the width of the pathway? Answer: (5x + 3) Solution: Area of a rectangle = length * width, so width = area / length. Width = (15x² + 34x + 15) / (3x + 5). Factor the numerator 15x² + 34x + 15.
    Full step-by-step solution

    Step 1: Area of a rectangle = length * width, so width = area / length. Step 2: Width = (15x² + 34x + 15) / (3x + 5). Step 3: Factor the numerator 15x² + 34x + 15. Step 4: Multiply the coefficient of x² (15) by the constant term (15): 15 * 15 = 225. Step 5: Find two numbers that multiply to 225 and add to 34. The numbers are 25 and 9 (since 25 * 9 = 225 and 25 + 9 = 34). Step 6: Rewrite the middle term using these numbers: 15x² + 25x + 9x + 15. Step 7: Group the terms: (15x² + 25x) + (9x + 15). Step 8: Factor each group: 5x(3x + 5) + 3(3x + 5). Step 9: Factor out the common binomial (3x + 5): (3x + 5)(5x + 3). Step 10: Width = (3x + 5)(5x + 3) / (3x + 5) = 5x + 3. The width of the pathway is (5x + 3) meters.

  5. x² - 9x + 20 = ? Answer: (x-4)(x-5) Solution: Identify coefficients: a=1, b=-9, c=20 Find two numbers that multiply to 20 and add to -9 The numbers are -4 and -5 because (-4)×(-5)=20 and (-4)+(-5)=-9 Write the factored form: (x-4)(x-5) The answer is (x-4)(x-5).
    Full step-by-step solution

    Step 1: Identify coefficients: a=1, b=-9, c=20 Step 2: Find two numbers that multiply to 20 and add to -9 Step 3: The numbers are -4 and -5 because (-4)×(-5)=20 and (-4)+(-5)=-9 Step 4: Write the factored form: (x-4)(x-5) The answer is (x-4)(x-5).

  6. Factor completely: 7x² - 27x + 18 = ? Answer: (7x - 6)(x - 3) Solution: Multiply the leading coefficient (7) by the constant term (18): 7 × 18 = 126 Find two numbers that multiply to 126 and add to -27: -21 and -6 (since -21 × -6 = 126 and -21 + -6 = -27) Rewrite the middle term using these numbers: 7x² - 21x - 6x + 18 Factor by grouping: (7x² - 21x) + (-6x + 18)…
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (7) by the constant term (18): 7 × 18 = 126 Step 2: Find two numbers that multiply to 126 and add to -27: -21 and -6 (since -21 × -6 = 126 and -21 + -6 = -27) Step 3: Rewrite the middle term using these numbers: 7x² - 21x - 6x + 18 Step 4: Factor by grouping: (7x² - 21x) + (-6x + 18) Step 5: Factor out common factors: 7x(x - 3) - 6(x - 3) Step 6: Factor out the common binomial: (7x - 6)(x - 3) The completely factored form is (7x - 6)(x - 3).

  7. Factor completely: 12x² + 37x + 21 = ? Answer: (3x + 7)(4x + 3) Solution: Multiply the leading coefficient (12) by the constant term (21): 12 × 21 = 252. Find two numbers that multiply to 252 and add to 37. The numbers are 28 and 9 because 28 × 9 = 252 and 28 + 9 = 37.
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (12) by the constant term (21): 12 × 21 = 252. Step 2: Find two numbers that multiply to 252 and add to 37. The numbers are 28 and 9 because 28 × 9 = 252 and 28 + 9 = 37. Step 3: Rewrite the middle term 37x as 28x + 9x: 12x² + 28x + 9x + 21. Step 4: Factor by grouping: (12x² + 28x) + (9x + 21). Step 5: Factor out the greatest common factor from each group: 4x(3x + 7) + 3(3x + 7). Step 6: Factor out the common binomial (3x + 7): (3x + 7)(4x + 3). The completely factored form is (3x + 7)(4x + 3).

  8. Factor completely: 14x² - 41x + 15 = ? Answer: (7x - 3)(2x - 5) Solution: Multiply the leading coefficient (14) by the constant term (15): 14 × 15 = 210 Find two numbers that multiply to 210 and add to -41: -35 and -6 (since -35 × -6 = 210 and -35 + -6 = -41) Rewrite the middle term using these numbers: 14x² - 35x - 6x + 15 Factor by grouping: (14x² - 35x) + (-6x +…
    Full step-by-step solution

    Step 1: Multiply the leading coefficient (14) by the constant term (15): 14 × 15 = 210 Step 2: Find two numbers that multiply to 210 and add to -41: -35 and -6 (since -35 × -6 = 210 and -35 + -6 = -41) Step 3: Rewrite the middle term using these numbers: 14x² - 35x - 6x + 15 Step 4: Factor by grouping: (14x² - 35x) + (-6x + 15) Step 5: Factor out common factors: 7x(2x - 5) - 3(2x - 5) Step 6: Factor out the common binomial: (7x - 3)(2x - 5) The completely factored form is (7x - 3)(2x - 5).