Factor Difference Squares

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Answer Key & Explanations

Factor Difference Squares · Grade 9 · Worksheet 2

1. Emma is designing a square mosaic tile for an art project. The tile is composed of a smaller square of colored glass placed at the center of a larger square frame made of metal. The side length of the entire mosaic (frame plus glass) is (7x + 3) centimeters. The side length of the central glass square is (7x - 3) centimeters. Using the difference of squares, what is the area of just the metal frame in square centimeters? Answer: 84x Solution: The area of the entire mosaic (outer square) is (7x + 3)^2. The area of the glass square (inner square) is (7x - 3)^2. The area of the metal frame is the difference: (7x + 3)^2 - (7x - 3)^2.
   Full step-by-step solution

   Step 1: The area of the entire mosaic (outer square) is (7x + 3)^2. Step 2: The area of the glass square (inner square) is (7x - 3)^2. Step 3: The area of the metal frame is the difference: (7x + 3)^2 - (7x - 3)^2. Step 4: Recognize this as a difference of squares: a^2 - b^2, where a = (7x + 3) and b = (7x - 3). Step 5: Apply the formula a^2 - b^2 = (a + b)(a - b): a + b = (7x + 3) + (7x - 3) = 14x a - b = (7x + 3) - (7x - 3) = 7x + 3 - 7x + 3 = 6 Step 6: Multiply: 14x * 6 = 84x The area of the metal frame is 84x square centimeters.

2. Aisha is designing a solar panel installation for her school's rooftop. The total area available for panels can be expressed as 49x² - 64 square meters. She realizes this area represents the difference between the area of the entire rectangular roof section and the area of a ventilation shaft that needs to remain uncovered. If the length of the roof section is (7x + 8) meters, what expression represents the width of the available installation area? Answer: (7x - 8) Solution: The difference of squares is a special factoring pattern where a binomial of the form a² - b² can be factored as (a + b)(a - b). The key is identifying what expressions represent 'a' and 'b' in the given scenario.
   Full step-by-step solution

   The difference of squares is a special factoring pattern where a binomial of the form a² - b² can be factored as (a + b)(a - b). This pattern appears frequently in real-world applications involving areas, such as when calculating remaining space after removing a smaller area from a larger one. The key is identifying what expressions represent 'a' and 'b' in the given scenario.

3. Mason is designing a rectangular garden in his backyard. The area of the garden is represented by the expression 121x² - 100 square feet. He knows that the length of the garden is (11x + 10) feet. If the area of a rectangle is equal to its length times its width, what expression represents the width of Mason's garden in terms of x? Answer: (11x - 10) Solution: The area of the garden is given as 121x² - 100 square feet. The length is given as (11x + 10) feet. Since area = length × width, we have width = area ÷ length.
   Full step-by-step solution

   Step 1: The area of the garden is given as 121x² - 100 square feet. Step 2: The length is given as (11x + 10) feet. Step 3: Since area = length × width, we have width = area ÷ length. Step 4: So width = (121x² - 100) ÷ (11x + 10). Step 5: Factor the numerator as a difference of squares: 121x² - 100 = (11x)² - (10)² = (11x + 10)(11x - 10). Step 6: Substitute back: width = [(11x + 10)(11x - 10)] ÷ (11x + 10). Step 7: Cancel the common factor (11x + 10) to get width = 11x - 10. Step 8: The width of Mason's garden is (11x - 10) feet.

4. Liam is designing a rectangular garden with an area of 121 square meters. He wants to create a decorative stone border along the perimeter, but needs to calculate the side lengths first. If the garden's area can be expressed as the difference of two perfect squares, x² - y², and the longer side is 2 meters more than the shorter side, what are the dimensions of Liam's garden? Answer: 11 meters by 11 meters 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 garden design, this pattern often appears when dealing with areas. When you have a perfect square area and know the relationship between the two dimensions, you can use algebraic reasoning to determine the exact measurements. This method is commonly used in architecture and landscaping to calculate dimensions from given areas.

5. Aisha is designing a solar panel array for her school's science fair project. The total area available for the panels can be expressed as (49x² - 36) square meters. She realizes this area represents the difference between the area of the entire roof space and the area of a ventilation shaft. If the length of the available space is (7x + 6) meters, what expression represents the width of the available space in terms of x? Answer: (7x - 6) Solution: The area is given as 49x² - 36, which is a difference of squares. Recognize that 49x² = (7x)² and 36 = 6². So 49x² - 36 = (7x + 6)(7x - 6).
   Full step-by-step solution

   Step 1: The area is given as 49x² - 36, which is a difference of squares. Step 2: Recognize that 49x² = (7x)² and 36 = 6². Step 3: Apply the difference of squares formula: a² - b² = (a + b)(a - b), where a = 7x and b = 6. Step 4: So 49x² - 36 = (7x + 6)(7x - 6). Step 5: The length is given as (7x + 6) meters. Step 6: Since area = length × width, we can write: (7x + 6)(width) = (7x + 6)(7x - 6). Step 7: Divide both sides by (7x + 6) to find the width: width = (7x - 6). The width of the available space is (7x - 6) meters.

6. Liam is designing a rectangular garden with an area of (x² - 16) square meters. He wants to build a decorative stone border along the perimeter. 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 Width = unknown (call it 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 The problem says: Area = x² - 16 Length = x + 4 Width = unknown (call it 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 factored form into the area 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): (x - 4)(x + 4) / (x + 4) = W Step 4: Cancel (x + 4) from numerator and denominator on the left side: x - 4 = W So the width is (x - 4) meters. Final answer: (x - 4)