Factor Difference Squares

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Answer Key & Explanations

Factor Difference Squares · Grade 9 · Worksheet 1

1. x⁴ - 81 = ? Answer: (x² + 9)(x + 3)(x - 3) Solution: Recognize that x⁴ - 81 is a difference of squares: (x²)² - (9)² Apply the difference of squares formula: a² - b² = (a + b)(a - b) Substitute a = x² and b = 9: (x² + 9)(x² - 9) Notice that x² - 9 is also a difference of squares: (x)² - (3)² Apply the difference of squares formula again: (x + 3)(x…
   Full step-by-step solution

   Step 1: Recognize that x⁴ - 81 is a difference of squares: (x²)² - (9)² Step 2: Apply the difference of squares formula: a² - b² = (a + b)(a - b) Step 3: Substitute a = x² and b = 9: (x² + 9)(x² - 9) Step 4: Notice that x² - 9 is also a difference of squares: (x)² - (3)² Step 5: Apply the difference of squares formula again: (x + 3)(x - 3) Step 6: Combine all factors: (x² + 9)(x + 3)(x - 3) The fully factored form is (x² + 9)(x + 3)(x - 3).

2. A square painting is mounted on a square frame, creating a border of uniform width around the painting. The entire artwork (painting plus frame) has side length (x + 6) inches, while the painting itself has side length (x - 6) inches. Using the difference of squares, what is the area of just the frame in square inches? Answer: 24x Solution: Find the area of the entire artwork (painting plus frame) Area of entire artwork = (x + 6)² Area of painting = (x - 6)² Frame area = (x + 6)² - (x - 6)² Apply the difference of squares formula: a² - b² = (a + b)(a - b) Where a = (x + 6) and b = (x - 6) Calculate (a + b) = (x + 6) + (x - 6) = 2x…
   Full step-by-step solution

   Step 1: Find the area of the entire artwork (painting plus frame) Area of entire artwork = (x + 6)² Step 2: Find the area of just the painting Area of painting = (x - 6)² Step 3: Find the area of just the frame by subtracting Frame area = (x + 6)² - (x - 6)² Step 4: Apply the difference of squares formula: a² - b² = (a + b)(a - b) Where a = (x + 6) and b = (x - 6) Step 5: Calculate (a + b) = (x + 6) + (x - 6) = 2x Step 6: Calculate (a - b) = (x + 6) - (x - 6) = 12 Step 7: Multiply the results Frame area = (2x)(12) = 24x The area of just the frame is 24x square inches.

3. A construction company is designing a square plaza with a smaller square fountain in the center. The total area available for the plaza is (9x² - 16) square meters. The fountain will occupy an area that can be expressed as a perfect square. If the side length of the fountain is (3x - 4) meters, what is the side length of the entire plaza in terms of x? Answer: 3x + 4 Solution: The total plaza area is 9x² - 16, which represents the difference between the plaza area and fountain area. The fountain area is (3x - 4)² = 9x² - 24x + 16. However, we need to find the plaza side length directly.
   Full step-by-step solution

   Step 1: The total plaza area is 9x² - 16, which represents the difference between the plaza area and fountain area. Step 2: The fountain area is (3x - 4)² = 9x² - 24x + 16. Step 3: However, we need to find the plaza side length directly. The total area 9x² - 16 is a difference of squares. Step 4: Factor 9x² - 16 = (3x)² - (4)² = (3x + 4)(3x - 4). Step 5: Since the fountain has side length (3x - 4), the plaza must have side length (3x + 4). Step 6: Verify: Plaza area = (3x + 4)(3x - 4) = 9x² - 16, which matches the given total area. The answer is 3x + 4.

4. Aisha is designing a solar panel array that will cover an area of (49x² - 64) square meters. She realizes this area represents the difference between the total available space and the area needed for maintenance access. If the length of the array is (7x + 8) meters, what expression represents the width of the solar panel array in terms of x? Answer: (7x - 8) Solution: The area of the solar panel array is given as (49x² - 64) square meters. The length is given as (7x + 8) meters.
   Full step-by-step solution

   Step 1: The area of the solar panel array is given as (49x² - 64) square meters. Step 2: The length is given as (7x + 8) meters. Step 3: Since area = length × width, we can write: width = area ÷ length Step 4: So width = (49x² - 64) ÷ (7x + 8) Step 5: Factor the numerator as a difference of squares: 49x² - 64 = (7x)² - 8² = (7x + 8)(7x - 8) Step 6: Now substitute back: width = [(7x + 8)(7x - 8)] ÷ (7x + 8) Step 7: Cancel the common factor (7x + 8): width = 7x - 8 Step 8: The width of the solar panel array is (7x - 8) meters.

5. A rectangular solar panel has an area represented by the expression 25x² - 49y² square meters. The length of the panel is (5x + 7y) meters. What expression represents the width of the solar panel? Answer: 5x - 7y Solution: The area of a rectangle equals length times width, so width = area ÷ length Width = (25x² - 49y²) ÷ (5x + 7y) Recognize that 25x² - 49y² is a difference of squares: (5x)² - (7y)² Factor the difference of squares: (5x)² - (7y)² = (5x + 7y)(5x - 7y) Substitute back: Width = [(5x + 7y)(5x - 7y)] ÷…
   Full step-by-step solution

   Step 1: The area of a rectangle equals length times width, so width = area ÷ length Step 2: Width = (25x² - 49y²) ÷ (5x + 7y) Step 3: Recognize that 25x² - 49y² is a difference of squares: (5x)² - (7y)² Step 4: Factor the difference of squares: (5x)² - (7y)² = (5x + 7y)(5x - 7y) Step 5: Substitute back: Width = [(5x + 7y)(5x - 7y)] ÷ (5x + 7y) Step 6: Cancel the common factor (5x + 7y) Step 7: Width = 5x - 7y The answer is 5x - 7y.

6. 81x² - 64y² = ? Answer: (9x - 8y)(9x + 8y) Solution: Identify the pattern as a difference of squares: a² - b² Recognize that 81x² = (9x)² and 64y² = (8y)² Apply the difference of squares formula: a² - b² = (a - b)(a + b) Substitute a = 9x and b = 8y Write the factored form: (9x - 8y)(9x + 8y) The answer is (9x - 8y)(9x + 8y).
   Full step-by-step solution

   Step 1: Identify the pattern as a difference of squares: a² - b² Step 2: Recognize that 81x² = (9x)² and 64y² = (8y)² Step 3: Apply the difference of squares formula: a² - b² = (a - b)(a + b) Step 4: Substitute a = 9x and b = 8y Step 5: Write the factored form: (9x - 8y)(9x + 8y) The answer is (9x - 8y)(9x + 8y).

7. 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.