Factoring Quadratic Expressions

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Answer Key & Explanations

Factoring Quadratic Expressions · Grade 10 · Worksheet 2

1. 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 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** \( 65 = 2w^2 + 3w \) Subtract 65 from both sides: \( 2w^2 + 3w - 65 = 0 \) --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here \( a = 2 \), \( b = 3 \), \( c = -65 \). Discriminant: \( D = b^2 - 4ac = 9 - 4(2)(-65) = 9 + 520 = 529 \). Square root of 529: \( \sqrt{529} = 23 \). So: \( w = \frac{-3 \pm 23}{4} \) --- **Step 5: Find possible values for w** First: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Second: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) Width cannot be negative, so \( w = 5 \). --- **Step 6: Find length** \( l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \) --- **Step 7: Final answer** Width = 5 meters, Length = 13 meters. --- **Step 8: Check** Area = \( 5 \times 13 = 65 \) square meters. Length (13) is indeed 3 more than twice the width (2×5 = 10, plus 3 = 13). Correct.

2. 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).

3. A rectangular garden has an area that can be modeled by the quadratic expression 2x² + 7x + 6 square meters, where x represents the width in meters. If the length is 5 meters longer than the width, determine the actual dimensions of the garden. Answer: width = 2 meters, length = 7 meters Solution: In geometry problems involving rectangular areas, the area equals length times width.
   Full step-by-step solution

   In geometry problems involving rectangular areas, the area equals length times width. When given a quadratic expression for area and a relationship between dimensions, you can factor the quadratic to find possible dimension pairs. The correct pair will satisfy both the area equation and the given relationship between length and width.

4. 2x² - 11x + 12 = 0 Answer: x = 4, 1.5 Solution: Identify coefficients: a = 2, b = -11, c = 12 Multiply a × c = 2 × 12 = 24 Find two numbers that multiply to 24 and add to -11: -8 and -3 Rewrite the middle term: 2x² - 8x - 3x + 12 = 0 Factor by grouping: (2x² - 8x) + (-3x + 12) = 0 Factor out common terms: 2x(x - 4) - 3(x - 4) = 0 Factor out…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 2, b = -11, c = 12 Step 2: Multiply a × c = 2 × 12 = 24 Step 3: Find two numbers that multiply to 24 and add to -11: -8 and -3 Step 4: Rewrite the middle term: 2x² - 8x - 3x + 12 = 0 Step 5: Factor by grouping: (2x² - 8x) + (-3x + 12) = 0 Step 6: Factor out common terms: 2x(x - 4) - 3(x - 4) = 0 Step 7: Factor out (x - 4): (x - 4)(2x - 3) = 0 Step 8: Set each factor equal to zero: x - 4 = 0 or 2x - 3 = 0 Step 9: Solve each equation: x = 4 or x = 3/2 = 1.5 The solutions are x = 4 and x = 1.5.

5. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (x,0,0), (0,2x,0), and (0,0,3). The volume of the prism is 150 cubic units. Find the value of x. Answer: 5 Solution: Identify the dimensions from the coordinates. The prism has vertices at (0,0,0), (x,0,0), (0,2x,0), and (0,0,3). The distance from (0,0,0) to (x,0,0) gives the length = x.
   Full step-by-step solution

   Step 1: Identify the dimensions from the coordinates. The prism has vertices at (0,0,0), (x,0,0), (0,2x,0), and (0,0,3). Step 2: The distance from (0,0,0) to (x,0,0) gives the length = x. Step 3: The distance from (0,0,0) to (0,2x,0) gives the width = 2x. Step 4: The distance from (0,0,0) to (0,0,3) gives the height = 3. Step 5: Volume of a rectangular prism = length × width × height = x × 2x × 3 = 6x². Step 6: Set the volume equal to 150: 6x² = 150. Step 7: Divide both sides by 6: x² = 25. Step 8: Take the square root: x = 5 (since length must be positive). The answer is 5.

6. Factor: 9x² - 49 Answer: (3x - 7)(3x + 7) Solution: Recognize the expression as a difference of squares: 9x² - 49 = (3x)² - 7². Write the factors: (3x - 7)(3x + 7).
   Full step-by-step solution

   Step 1: Recognize the expression as a difference of squares: 9x² - 49 = (3x)² - 7². Step 2: Apply the formula a² - b² = (a - b)(a + b), where a = 3x and b = 7. Step 3: Write the factors: (3x - 7)(3x + 7). The answer is (3x - 7)(3x + 7).

7. Olivia is designing a rectangular tiled patio. The area of the patio in square feet is given by the quadratic expression 3x² + 17x + 10, where x is a positive integer representing the width in feet. If the length of the patio is (3x + 2) feet, what expression represents the width of the patio in terms of x? Answer: (x + 5) Solution: Area of a rectangle = length × width. We know area = 3x² + 17x + 10 and length = (3x + 2). So width = area / length = (3x² + 17x + 10) / (3x + 2).
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width. We know area = 3x² + 17x + 10 and length = (3x + 2). So width = area / length = (3x² + 17x + 10) / (3x + 2). Step 2: Factor the quadratic 3x² + 17x + 10. We need two numbers that multiply to 3 × 10 = 30 and add to 17. These numbers are 15 and 2. Step 3: Rewrite the middle term: 3x² + 15x + 2x + 10. Step 4: Group: (3x² + 15x) + (2x + 10). Step 5: Factor each group: 3x(x + 5) + 2(x + 5). Step 6: Factor out the common binomial (x + 5): (x + 5)(3x + 2). Step 7: So 3x² + 17x + 10 = (3x + 2)(x + 5). Since length = (3x + 2), the width must be (x + 5). The answer is (x + 5).

8. 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.