Factor Difference Squares

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Answer Key & Explanations

Factor Difference Squares · Grade 9 · Worksheet 1

1. A construction company is designing a square plaza with a smaller square fountain removed from one corner. The remaining area for walking space is given by the expression 49x² - 64 square meters. The project manager needs to determine the dimensions of this L-shaped walking area to calculate how much paving material to order. If the longer side of the rectangular walking section measures (7x + 8) meters, what is the width of this rectangular section in terms of x? Answer: 7x - 8 Solution: The area is given as 49x² - 64, which is a difference of squares Recognize that 49x² = (7x)² and 64 = 8² Factor using the difference of squares formula: a² - b² = (a + b)(a - b) Apply the formula: 49x² - 64 = (7x)² - 8² = (7x + 8)(7x - 8) The problem states the longer side is (7x + 8) meters…
   Full step-by-step solution

   Step 1: The area is given as 49x² - 64, which is a difference of squares Step 2: Recognize that 49x² = (7x)² and 64 = 8² Step 3: Factor using the difference of squares formula: a² - b² = (a + b)(a - b) Step 4: Apply the formula: 49x² - 64 = (7x)² - 8² = (7x + 8)(7x - 8) Step 5: The problem states the longer side is (7x + 8) meters Step 6: Therefore, the width must be (7x - 8) meters Step 7: The answer is 7x - 8

2. Liam is designing a rectangular garden with an area that can be expressed as (x² - 16) square meters. He realizes this area represents the difference between the area of the entire garden space and the area of a smaller square patio he plans to remove. If the garden's length is (x + 4) meters, what is its width in terms of x? Answer: (x - 4) Solution: - Area of the garden = \( x^2 - 16 \) square meters. - This area is the difference between the area of the entire garden space and the area of a smaller square patio removed.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We are told: - Area of the garden = \( x^2 - 16 \) square meters. - This area is the difference between the area of the entire garden space and the area of a smaller square patio removed. But actually, that detail is just to explain why the area is a "difference of squares" — we don't need it for the algebra. - The garden is a rectangle with length = \( x + 4 \) meters. - We need the width in terms of \( x \). --- **Step 2: Recall rectangle area formula** Area = length × width So: \[ x^2 - 16 = (x + 4) \times \text{width} \] --- **Step 3: Solve for width** Divide both sides by \( x + 4 \): \[ \text{width} = \frac{x^2 - 16}{x + 4} \] --- **Step 4: Factor the numerator** We know \( x^2 - 16 \) is a difference of squares: \[ x^2 - 16 = (x - 4)(x + 4) \] --- **Step 5: Simplify the fraction** \[ \text{width} = \frac{(x - 4)(x + 4)}{x + 4} \] Cancel \( x + 4 \) (valid as long as \( x \neq -4 \), which is fine for a physical length): \[ \text{width} = x - 4 \] --- **Step 6: Conclusion** The width of the garden is \( x - 4 \) meters. --- **Final answer:** \( x - 4 \)

3. A physics class is designing a solar panel array where the total area can be expressed as 49x² - 64 square meters. The teacher explains that this represents the difference between the area of the entire mounting surface and the area of a smaller control unit that will be removed. If the array's length is (7x + 8) meters, what expression represents the width of the solar panel array? Answer: (7x - 8) Solution: The difference of squares is a special factoring pattern where a² - b² factors as (a + b)(a - b).
   Full step-by-step solution

   The difference of squares is a special factoring pattern where a² - b² factors as (a + b)(a - b). This pattern appears frequently in algebra when dealing with areas, distances, or other measurements that can be expressed as the difference between two perfect squares. For example, if you had an area of 25y² - 36, you could factor it as (5y + 6)(5y - 6), where 5y comes from the square root of 25y² and 6 comes from the square root of 36.

4. 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.

5. A square garden has an area of 64x² - 81 square meters. If the side length of the garden is expressed as a binomial in terms of x, what is the length of one side of the garden? Answer: 8x - 9 Solution: The area of a square is given by side length squared. So if the area is 64x² - 81, then side² = 64x² - 81. Recognize that 64x² - 81 is a difference of squares.
   Full step-by-step solution

   Step 1: The area of a square is given by side length squared. So if the area is 64x² - 81, then side² = 64x² - 81. Step 2: Recognize that 64x² - 81 is a difference of squares. 64x² is (8x)² and 81 is 9². Step 3: Factor the difference of squares: a² - b² = (a - b)(a + b). Here a = 8x and b = 9. Step 4: So 64x² - 81 = (8x - 9)(8x + 9). Step 5: Since this represents the area of a square, and area = side × side, the side length must be one of these factors. Typically, we take the positive expression, so side length = 8x - 9 meters (assuming x is large enough to make this positive). The answer is 8x - 9.

6. 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).

7. Matiu is designing a rectangular garden where the area can be expressed as (36x² - 121) square meters. He wants to install a path that runs along the entire length of one side, which measures (6x + 11) meters. If the area of the garden is the product of its length and width, what expression represents the width of the garden in meters? Answer: (6x - 11) Solution: The area is given as 36x² - 121 square meters. Recognize this as a difference of squares: 36x² = (6x)² and 121 = 11². Write the expression as (6x)² - 11².
   Full step-by-step solution

   Step 1: The area is given as 36x² - 121 square meters. Step 2: Recognize this as a difference of squares: 36x² = (6x)² and 121 = 11². Step 3: Write the expression as (6x)² - 11². Step 4: Factor using the formula a² - b² = (a + b)(a - b): (6x + 11)(6x - 11). Step 5: The length is given as (6x + 11) meters. Step 6: Since area = length × width, width = area / length = (6x + 11)(6x - 11) / (6x + 11). Step 7: Cancel the common factor (6x + 11) to get width = (6x - 11) meters. The answer is (6x - 11).