Factor Difference Squares

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Answer Key & Explanations

Factor Difference Squares · Grade 9 · Worksheet 2

1. x² - 81 = ? Answer: (x - 9)(x + 9) Solution: Identify the pattern as a difference of squares: a² - b² Recognize that x² is (x)² and 81 is (9)² Apply the difference of squares formula: a² - b² = (a - b)(a + b) Substitute x for a and 9 for b: (x - 9)(x + 9) The answer is (x - 9)(x + 9).
   Full step-by-step solution

   Step 1: Identify the pattern as a difference of squares: a² - b² Step 2: Recognize that x² is (x)² and 81 is (9)² Step 3: Apply the difference of squares formula: a² - b² = (a - b)(a + b) Step 4: Substitute x for a and 9 for b: (x - 9)(x + 9) The answer is (x - 9)(x + 9).

2. A rectangular garden has an area represented by the expression x² - 16 square meters. If the length of the garden is (x + 4) meters, what is the width of the garden in terms of x? Answer: x - 4 Solution: Area = length × width Area = x² - 16 Length = x + 4 Let width = W. (x + 4) × W = x² - 16 Solve for W. W = (x² - 16) / (x + 4) Notice that x² - 16 is a difference of squares.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width Given: Area = x² - 16 Length = x + 4 Let width = W. So: (x + 4) × W = x² - 16 Step 1: Solve for W. W = (x² - 16) / (x + 4) Step 2: Notice that x² - 16 is a difference of squares. x² - 16 = x² - 4² = (x - 4)(x + 4) Step 3: Substitute this into the expression for W: W = [(x - 4)(x + 4)] / (x + 4) Step 4: Cancel the common factor (x + 4) from numerator and denominator (x ≠ -4, but for algebraic simplification it's allowed): W = x - 4 Thus, the width of the garden is x - 4 meters.

3. Liam is designing a rectangular garden with an area that can be expressed as (16x² - 81) square feet. He realizes this area represents a difference of two perfect squares. What are the dimensions (length and width) of Liam's garden, expressed as binomial factors? Answer: (4x - 9)(4x + 9) Solution: Identify the form of the problem. We are told the area is 16x² - 81, and it is a difference of two perfect squares. The general formula for a difference of squares is: a² - b² = (a - b)(a + b).
   Full step-by-step solution

   Step 1: Identify the form of the problem. We are told the area is 16x² - 81, and it is a difference of two perfect squares. The general formula for a difference of squares is: a² - b² = (a - b)(a + b). Step 2: Recognize the perfect squares in 16x² - 81. 16x² is a perfect square because (4x)² = 16x². 81 is a perfect square because 9² = 81. Step 3: Rewrite the expression as a difference of squares. 16x² - 81 = (4x)² - (9)². Step 4: Apply the difference of squares formula. Here, a = 4x and b = 9. So (4x)² - (9)² = (4x - 9)(4x + 9). Step 5: Interpret the result. The dimensions of the rectangular garden are the two binomial factors: Length = 4x + 9 feet, Width = 4x - 9 feet (or vice versa, since multiplication is commutative). Final answer: (4x - 9)(4x + 9)

4. Liam is designing a rectangular garden with an area of (x² - 25) square meters. He wants to install a decorative border around the perimeter. If the length of the garden is (x + 5) meters, what is the width of the garden in terms of x? Answer: (x - 5) Solution: Area = length × width Area = x² - 25 Length = x + 5 Width = unknown (call it W) (x + 5) × W = x² - 25 Recognize that x² - 25 is a difference of squares.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width The problem gives: Area = x² - 25 Length = x + 5 Width = unknown (call it W) So: (x + 5) × W = x² - 25 Step 1: Recognize that x² - 25 is a difference of squares. Difference of squares formula: a² - b² = (a - b)(a + b) Here, x² - 25 = x² - 5² = (x - 5)(x + 5) Step 2: Substitute the factored form into the area equation: (x + 5) × W = (x - 5)(x + 5) Step 3: Since x + 5 is a common factor on both sides (and assuming x + 5 ≠ 0), we can divide both sides by (x + 5): W = (x - 5)(x + 5) / (x + 5) Step 4: Cancel (x + 5) from numerator and denominator: W = x - 5 Thus, the width is (x - 5) meters.

5. Liam is designing a rectangular garden with an area of (x² - 25) square meters. He wants to build a fence around the perimeter. If the length of the garden is (x + 5) meters, what is the width of the garden in terms of x? Answer: (x - 5) Solution: Area = length × width Area = x² - 25 Length = x + 5 Width = unknown (call it W) x² - 25 = (x + 5) × W Recognize that x² - 25 is a difference of squares.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width The problem says: Area = x² - 25 Length = x + 5 Width = unknown (call it W) So: x² - 25 = (x + 5) × W Step 1: Recognize that x² - 25 is a difference of squares. Difference of squares formula: a² - b² = (a - b)(a + b) Here, x² - 25 = x² - 5² = (x - 5)(x + 5) Step 2: Substitute this into the area equation: (x - 5)(x + 5) = (x + 5) × W Step 3: Divide both sides by (x + 5) to solve for W. We can do this because x + 5 is not zero (since it's a length, it's positive). So: W = (x - 5)(x + 5) / (x + 5) Step 4: Cancel (x + 5) from numerator and denominator: W = x - 5 Thus, the width is (x - 5) meters. Final answer: x - 5

6. A construction company is designing a rectangular parking lot with an area of (4x² - 49) square meters. The project manager, Maria, needs to determine the dimensions for ordering materials. If the length of the parking lot is (2x + 7) meters, what expression represents the width of the parking lot in terms of x? Answer: (2x - 7) Solution: We know area = 4x² - 49 and length = 2x + 7 To find width, divide area by length: width = (4x² - 49) ÷ (2x + 7) Recognize that 4x² - 49 is a difference of squares: (2x)² - 7² Factor the difference of squares: (2x + 7)(2x - 7) Now width = [(2x + 7)(2x - 7)] ÷ (2x + 7) Cancel the common factor (2x…
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: We know area = 4x² - 49 and length = 2x + 7 Step 3: To find width, divide area by length: width = (4x² - 49) ÷ (2x + 7) Step 4: Recognize that 4x² - 49 is a difference of squares: (2x)² - 7² Step 5: Factor the difference of squares: (2x + 7)(2x - 7) Step 6: Now width = [(2x + 7)(2x - 7)] ÷ (2x + 7) Step 7: Cancel the common factor (2x + 7) Step 8: The width is (2x - 7) meters The answer is (2x - 7).

7. Aisha is designing a solar panel installation on a rectangular section of her roof. The total area available can be expressed as (49x² - 64) square feet. She realizes this represents the difference between the area of the entire roof section and the area where solar panels cannot be installed. If the length of the available section is (7x + 8) feet, what expression represents the width of the available installation area? Answer: (7x - 8) Solution: The difference of squares is a special factoring pattern where a² - b² = (a + b)(a - b).
   Full step-by-step solution

   The difference of squares is a special factoring pattern where a² - b² = (a + b)(a - b). In real-world applications like construction or design, this pattern often appears when working with areas. For example, if you have a rectangular area expressed as a difference of squares, the two factors represent the length and width of the rectangle.

8. A rectangular garden has an area of (x² - 16) square meters. If the length of the garden is (x + 4) meters, what is the width of the garden in terms of x? Answer: (x - 4) Solution: Area = length × width Area = x² - 16 Length = x + 4 Let the width be W. x² - 16 = (x + 4) × W Recognize that x² - 16 is a difference of squares.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width Given: Area = x² - 16 Length = x + 4 Let the width be W. So: x² - 16 = (x + 4) × W Step 1: Recognize that x² - 16 is a difference of squares. Difference of squares formula: a² - b² = (a - b)(a + b) Here, x² - 16 = x² - 4² = (x - 4)(x + 4) Step 2: Substitute this into the equation: (x - 4)(x + 4) = (x + 4) × W Step 3: Since x + 4 is a common factor on both sides (and assuming x ≠ -4 so we don't divide by zero), we can divide both sides by (x + 4): W = (x - 4)(x + 4) / (x + 4) Step 4: Cancel (x + 4) from numerator and denominator: W = x - 4 Thus, the width of the garden is (x - 4) meters.