Factoring Quadratic Expressions

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Factoring Quadratic Expressions · Grade 10 · Worksheet 1

1. Noah is building a rectangular sandbox for his younger sister. The area of the sandbox in square feet can be modeled by the quadratic expression 6x² + 19x + 10, where x is a positive integer representing the width in feet. If the length of the sandbox is (2x + 5) feet, what expression represents the width of the sandbox in terms of x? Answer: (3x + 2) Solution: The area of a rectangle is length times width. We know the area is 6x² + 19x + 10 and the length is (2x + 5). We need to find the width w such that (2x + 5) * w = 6x² + 19x + 10.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. We know the area is 6x² + 19x + 10 and the length is (2x + 5). We need to find the width w such that (2x + 5) * w = 6x² + 19x + 10. Step 2: Factor the quadratic 6x² + 19x + 10 using the grouping method. Multiply the leading coefficient (6) by the constant term (10): 6 * 10 = 60. Step 3: Find two numbers that multiply to 60 and add to 19. These numbers are 4 and 15 because 4 * 15 = 60 and 4 + 15 = 19. Step 4: Rewrite the middle term using these numbers: 6x² + 4x + 15x + 10. Step 5: Group the terms: (6x² + 4x) + (15x + 10). Step 6: Factor out the greatest common factor from each group: 2x(3x + 2) + 5(3x + 2). Step 7: Factor out the common binomial factor (3x + 2): (3x + 2)(2x + 5). Step 8: So the area factors as (2x + 5)(3x + 2). Since the length is (2x + 5), the width must be (3x + 2). The answer is (3x + 2).

2. A rectangular tile pattern is drawn on a coordinate plane. The tile's length is (x + 5) units and its width is (2x + 3) units. The area of the tile is expressed as a quadratic polynomial in standard form. What is the coefficient of the x term in this quadratic? Answer: 13 Solution: The area of a rectangle is length times width. Area = (x + 5)(2x + 3) First: x * 2x = 2x^2 Outer: x * 3 = 3x Inner: 5 * 2x = 10x Last: 5 * 3 = 15 Combine like terms: 2x^2 + (3x + 10x) + 15 = 2x^2 + 13x + 15 The quadratic in standard form is 2x^2 + 13x + 15.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. Step 2: Area = (x + 5)(2x + 3) Step 3: Use the distributive property (FOIL method): First: x * 2x = 2x^2 Outer: x * 3 = 3x Inner: 5 * 2x = 10x Last: 5 * 3 = 15 Step 4: Combine like terms: 2x^2 + (3x + 10x) + 15 = 2x^2 + 13x + 15 Step 5: The quadratic in standard form is 2x^2 + 13x + 15. Step 6: The coefficient of the x term is 13. The answer is 13.

3. Factor completely: 6x² - 17x + 12 = ? Answer: (2x-3)(3x-4) Solution: Multiply the leading coefficient and constant term: 6 × 12 = 72 Find two numbers that multiply to 72 and add to -17: -8 and -9 Rewrite the middle term: 6x² - 8x - 9x + 12 Factor by grouping: (6x² - 8x) + (-9x + 12) Factor out common factors: 2x(3x - 4) - 3(3x - 4) Factor out the common binomial:…
   Full step-by-step solution

   Step 1: Multiply the leading coefficient and constant term: 6 × 12 = 72 Step 2: Find two numbers that multiply to 72 and add to -17: -8 and -9 Step 3: Rewrite the middle term: 6x² - 8x - 9x + 12 Step 4: Factor by grouping: (6x² - 8x) + (-9x + 12) Step 5: Factor out common factors: 2x(3x - 4) - 3(3x - 4) Step 6: Factor out the common binomial: (2x - 3)(3x - 4) The completely factored form is (2x-3)(3x-4).

4. Factor: 12x² + 37x + 21 Answer: (3x + 7)(4x + 3) Solution: Identify a = 12, b = 37, c = 21. Multiply a and c: 12 × 21 = 252. Find two numbers that multiply to 252 and add to 37.
   Full step-by-step solution

   Step 1: Identify a = 12, b = 37, c = 21. Step 2: Multiply a and c: 12 × 21 = 252. Step 3: 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 4: Rewrite the middle term: 12x² + 28x + 9x + 21. Step 5: Factor by grouping: (12x² + 28x) + (9x + 21). Step 6: Factor out the GCF from each group: 4x(3x + 7) + 3(3x + 7). Step 7: Factor out the common binomial (3x + 7): (3x + 7)(4x + 3). The factored form is (3x + 7)(4x + 3).

5. x² - 5x - 36 = 0 Answer: x = 9, -4 Solution: Identify the quadratic equation: x² - 5x - 36 = 0 Find two numbers that multiply to -36 and add to -5 The numbers are -9 and 4 because (-9) × 4 = -36 and (-9) + 4 = -5 Factor the quadratic: (x - 9)(x + 4) = 0 Set each factor equal to zero: x - 9 = 0 or x + 4 = 0 Solve each equation: x = 9 or x =…
   Full step-by-step solution

   Step 1: Identify the quadratic equation: x² - 5x - 36 = 0 Step 2: Find two numbers that multiply to -36 and add to -5 Step 3: The numbers are -9 and 4 because (-9) × 4 = -36 and (-9) + 4 = -5 Step 4: Factor the quadratic: (x - 9)(x + 4) = 0 Step 5: Set each factor equal to zero: x - 9 = 0 or x + 4 = 0 Step 6: Solve each equation: x = 9 or x = -4 The solutions are x = 9 and x = -4.

6. A company is designing a new smartphone with a rectangular screen. The screen's area can be modeled by the polynomial 3x² + 14x - 5 square centimeters, where x is a positive measurement. If the screen's length is (3x - 1) centimeters, what expression represents the width of the screen? Answer: (x + 5) Solution: In algebra, when working with polynomial areas of rectangles, the width can be found by dividing the area polynomial by the length polynomial.
   Full step-by-step solution

   In algebra, when working with polynomial areas of rectangles, the width can be found by dividing the area polynomial by the length polynomial. This is similar to factoring quadratics, where if you know one factor, you can find the other by polynomial division or by recognizing the factored form. For example, if a rectangle's area is x² + 5x + 6 and the length is (x + 2), the width would be (x + 3) since (x + 2)(x + 3) = x² + 5x + 6.

7. A right triangle has legs with lengths represented by the expressions (x + 3) and (x + 5). The area of the triangle is 84 square units. What is the positive value of x? Answer: 9 Solution: The area of a right triangle is (1/2) × leg1 × leg2 Substitute the given expressions: (1/2)(x + 3)(x + 5) = 84 Multiply both sides by 2: (x + 3)(x + 5) = 168 Expand the left side: x² + 5x + 3x + 15 = 168 Simplify: x² + 8x + 15 = 168 Subtract 168 from both sides: x² + 8x - 153 = 0 Factor the…
   Full step-by-step solution

   Step 1: The area of a right triangle is (1/2) × leg1 × leg2 Step 2: Substitute the given expressions: (1/2)(x + 3)(x + 5) = 84 Step 3: Multiply both sides by 2: (x + 3)(x + 5) = 168 Step 4: Expand the left side: x² + 5x + 3x + 15 = 168 Step 5: Simplify: x² + 8x + 15 = 168 Step 6: Subtract 168 from both sides: x² + 8x - 153 = 0 Step 7: Factor the quadratic: (x + 17)(x - 9) = 0 Step 8: Solve for x: x = -17 or x = 9 Step 9: Since length must be positive, x = 9 The answer is 9.

8. Sophia is designing a rectangular rose garden for her school's landscaping project. The area of the garden is given by the quadratic expression 6x² + 31x + 35 square feet, where x is a positive integer representing a scaling factor. The length of the garden is known to be (3x + 5) feet. What expression represents the width of the garden in feet? Answer: 2x + 7 Solution: Area of a rectangle = length × width, so width = area ÷ length. Area = 6x² + 31x + 35, length = 3x + 5. To find width, factor the quadratic 6x² + 31x + 35.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so width = area ÷ length. Step 2: Area = 6x² + 31x + 35, length = 3x + 5. Step 3: To find width, factor the quadratic 6x² + 31x + 35. Step 4: Look for two numbers that multiply to 6 × 35 = 210 and add to 31. The numbers are 10 and 21. Step 5: Rewrite the middle term: 6x² + 10x + 21x + 35. Step 6: Group the terms: (6x² + 10x) + (21x + 35). Step 7: Factor each group: 2x(3x + 5) + 7(3x + 5). Step 8: Factor out the common binomial: (3x + 5)(2x + 7). Step 9: Since length is (3x + 5), width must be (2x + 7). The answer is 2x + 7.