Factoring Quadratic Expressions

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Factoring Quadratic Expressions · Grade 10 · Worksheet 2

1. Emma is designing a rectangular mural on a coordinate grid. The four corners of the mural are at (0,0), (3x+1,0), (3x+1,2x-3), and (0,2x-3). The area of the mural is 130 square units. Factor the quadratic expression that represents the area in terms of x to find the binomial factors. What are those binomial factors? Answer: (3x+1)(2x-3) Solution: Identify the dimensions of the rectangle. The length is from x=0 to x=3x+1, so length = 3x+1. The width is from y=0 to y=2x-3, so width = 2x-3.
   Full step-by-step solution

   Step 1: Identify the dimensions of the rectangle. The length is from x=0 to x=3x+1, so length = 3x+1. The width is from y=0 to y=2x-3, so width = 2x-3. Step 2: The area of a rectangle is length times width, so Area = (3x+1)(2x-3). Step 3: Multiply the binomials using FOIL: First: 3x*2x = 6x^2. Outer: 3x*(-3) = -9x. Inner: 1*2x = 2x. Last: 1*(-3) = -3. Step 4: Combine like terms: 6x^2 + (-9x+2x) + (-3) = 6x^2 - 7x - 3. Step 5: The area is given as 130, so 6x^2 - 7x - 3 = 130. But the problem asks for the binomial factors of the area expression, not to solve for x. The area expression in factored form is already (3x+1)(2x-3). Step 6: The binomial factors are (3x+1) and (2x-3).

2. Factor completely: 6x² - 11x - 10 = ? Answer: (3x + 2)(2x - 5) Solution: Multiply the leading coefficient (6) by the constant term (-10): 6 × (-10) = -60 Find two numbers that multiply to -60 and add to -11: -15 and 4 Rewrite the middle term: 6x² - 15x + 4x - 10 Factor by grouping: (6x² - 15x) + (4x - 10) Factor out common factors: 3x(2x - 5) + 2(2x - 5) Factor out…
   Full step-by-step solution

   Step 1: Multiply the leading coefficient (6) by the constant term (-10): 6 × (-10) = -60 Step 2: Find two numbers that multiply to -60 and add to -11: -15 and 4 Step 3: Rewrite the middle term: 6x² - 15x + 4x - 10 Step 4: Factor by grouping: (6x² - 15x) + (4x - 10) Step 5: Factor out common factors: 3x(2x - 5) + 2(2x - 5) Step 6: Factor out the common binomial: (3x + 2)(2x - 5) The completely factored form is (3x + 2)(2x - 5).

3. Kaia is designing a rectangular solar panel for a school project. The area of the panel is given by the polynomial 6x² + 23x - 18 square meters, where x is a positive integer representing a scaling factor. If the length of the panel is (3x - 2) meters, write an expression for the width of the panel in terms of x. Answer: 2x + 9 Solution: The area of a rectangle is length times width. We know area = 6x² + 23x - 18 and length = 3x - 2. To find the width, divide the area by the length: width = (6x² + 23x - 18) / (3x - 2).
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. We know area = 6x² + 23x - 18 and length = 3x - 2. Step 2: To find the width, divide the area by the length: width = (6x² + 23x - 18) / (3x - 2). Step 3: Factor the numerator 6x² + 23x - 18. We need two numbers that multiply to 6 * (-18) = -108 and add to 23. The numbers are 27 and -4 because 27 * (-4) = -108 and 27 + (-4) = 23. Step 4: Rewrite the middle term: 6x² + 27x - 4x - 18. Step 5: Factor by grouping: 3x(2x + 9) - 2(2x + 9) = (2x + 9)(3x - 2). Step 6: So width = (2x + 9)(3x - 2) / (3x - 2) = 2x + 9. The answer is 2x + 9.

4. Factor: 8x² + 22x + 15 Answer: (2x + 3)(4x + 5) Solution: Identify a = 8, b = 22, c = 15. Multiply a and c: 8 × 15 = 120. Find two numbers that multiply to 120 and add to 22: 10 and 12.
   Full step-by-step solution

   Step 1: Identify a = 8, b = 22, c = 15. Step 2: Multiply a and c: 8 × 15 = 120. Step 3: Find two numbers that multiply to 120 and add to 22: 10 and 12. Step 4: Rewrite the middle term: 8x² + 10x + 12x + 15. Step 5: Factor by grouping: (8x² + 10x) + (12x + 15). Step 6: Factor out the GCF from each group: 2x(4x + 5) + 3(4x + 5). Step 7: Factor out the common binomial (4x + 5): (2x + 3)(4x + 5). The factored form is (2x + 3)(4x + 5).

5. Olivia is analyzing a rectangular garden plot drawn on a coordinate grid. One corner is at the origin, and the adjacent corners are at (3x+1, 0) and (0, x-3). The area of the rectangle is 42 square units. Express the area as a quadratic polynomial in standard form and then factor it completely. What are the binomial factors of the area expression? Answer: (3x+1)(x-3) Solution: The rectangle has one corner at (0,0). The adjacent corner on the x-axis is at (3x+1, 0), so the length along the x-axis is 3x+1. The adjacent corner on the y-axis is at (0, x-3), so the width along the y-axis is x-3.
   Full step-by-step solution

   Step 1: The rectangle has one corner at (0,0). The adjacent corner on the x-axis is at (3x+1, 0), so the length along the x-axis is 3x+1. The adjacent corner on the y-axis is at (0, x-3), so the width along the y-axis is x-3. Step 2: Area of a rectangle = length times width = (3x+1)(x-3). Step 3: Multiply using FOIL: First: 3x * x = 3x^2. Outer: 3x * (-3) = -9x. Inner: 1 * x = x. Last: 1 * (-3) = -3. Step 4: Combine like terms: 3x^2 + (-9x + x) + (-3) = 3x^2 - 8x - 3. Step 5: The area is given as 42, so the quadratic expression for area in standard form is 3x^2 - 8x - 3. Step 6: Factor 3x^2 - 8x - 3. Look for two numbers that multiply to 3 * (-3) = -9 and add to -8. Those numbers are -9 and 1. Step 7: Rewrite the middle term: 3x^2 - 9x + 1x - 3. Step 8: Factor by grouping: (3x^2 - 9x) + (1x - 3) = 3x(x - 3) + 1(x - 3). Step 9: Factor out the common binomial (x - 3): (x - 3)(3x + 1). Therefore, the binomial factors are (3x+1)(x-3).

6. Aroha is drawing a rectangle on a coordinate grid. The rectangle has one vertex at the origin (0, 0), and the opposite vertex at (x + 3, 2x - 1). The area of the rectangle is 30 square units. If the expression for the area, in factored form, is (x + 3)(2x - 1), what are the binomial factors of the area expression when it is written in standard quadratic form? Answer: (x + 3)(2x - 1) Solution: The area of the rectangle is given by (x + 3)(2x - 1). This is already the factored form. The problem asks for the binomial factors of the area expression when written in standard quadratic form.
   Full step-by-step solution

   Step 1: The area of the rectangle is given by (x + 3)(2x - 1). This is already the factored form. Step 2: The problem asks for the binomial factors of the area expression when written in standard quadratic form. The standard form is obtained by expanding: (x + 3)(2x - 1). Step 3: Expand using the distributive property: (x)(2x) + (x)(-1) + (3)(2x) + (3)(-1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3. Step 4: The standard quadratic form is 2x^2 + 5x - 3. Step 5: To factor this back into binomials, we look for two numbers that multiply to (2)(-3) = -6 and add to 5. Those numbers are 6 and -1. Step 6: Rewrite the middle term: 2x^2 + 6x - x - 3. Step 7: Factor by grouping: 2x(x + 3) - 1(x + 3) = (x + 3)(2x - 1). Step 8: The binomial factors are (x + 3) and (2x - 1). The answer is (x + 3)(2x - 1).

7. 2x² + 7x - 15 = 0 Answer: x = 3/2, -5 Solution: Identify coefficients: a = 2, b = 7, c = -15 Multiply a × c = 2 × (-15) = -30 Find two numbers that multiply to -30 and add to 7: 10 and -3 Rewrite the middle term: 2x² + 10x - 3x - 15 = 0 Factor by grouping: (2x² + 10x) + (-3x - 15) = 0 Factor out common terms: 2x(x + 5) - 3(x + 5) = 0 Factor…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 2, b = 7, c = -15 Step 2: Multiply a × c = 2 × (-15) = -30 Step 3: Find two numbers that multiply to -30 and add to 7: 10 and -3 Step 4: Rewrite the middle term: 2x² + 10x - 3x - 15 = 0 Step 5: Factor by grouping: (2x² + 10x) + (-3x - 15) = 0 Step 6: Factor out common terms: 2x(x + 5) - 3(x + 5) = 0 Step 7: Factor out (x + 5): (2x - 3)(x + 5) = 0 Step 8: Set each factor equal to zero: 2x - 3 = 0 or x + 5 = 0 Step 9: Solve each equation: x = 3/2 or x = -5 The solutions are x = 3/2 and x = -5.

8. Isabella is designing a rectangular garden on a coordinate grid. She places one corner at (0, 0) and the opposite corner at (2x + 1, x - 2). The area of the garden is 42 square units. Factor the quadratic expression representing the area to find the value of x. Answer: (2x + 1)(x - 2) Solution: The length of the rectangle is the x-coordinate of the opposite corner: 2x + 1. The width is the y-coordinate: x - 2. Area = length times width = (2x + 1)(x - 2).
   Full step-by-step solution

   Step 1: The length of the rectangle is the x-coordinate of the opposite corner: 2x + 1. The width is the y-coordinate: x - 2. Step 2: Area = length times width = (2x + 1)(x - 2). Step 3: Expand using FOIL: First: 2x * x = 2x^2. Outer: 2x * (-2) = -4x. Inner: 1 * x = x. Last: 1 * (-2) = -2. Step 4: Combine like terms: 2x^2 + (-4x + x) - 2 = 2x^2 - 3x - 2. Step 5: Set equal to 42: 2x^2 - 3x - 2 = 42. Subtract 42 from both sides: 2x^2 - 3x - 44 = 0. Step 6: Factor the quadratic: Look for factors of 2 * (-44) = -88 that add to -3. The factors are -11 and 8. Rewrite: 2x^2 - 11x + 8x - 44 = 0. Step 7: Group: (2x^2 - 11x) + (8x - 44) = 0. Factor each group: x(2x - 11) + 4(2x - 11) = 0. Step 8: Factor out (2x - 11): (2x - 11)(x + 4) = 0. Step 9: Solve for x: 2x - 11 = 0 gives x = 11/2 = 5.5; x + 4 = 0 gives x = -4 (reject since length must be positive). So x = 5.5. Step 10: The factored form of the area expression is (2x + 1)(x - 2).