Factor Difference Squares

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Factor Difference Squares · Grade 9 · Worksheet 3

1. 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.

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. Emma 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 space and the area of a ventilation shaft that will remain uncovered. If the length of the installation area is (7x + 8) meters, what expression represents the width of the installation area in terms of x? 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). In real-world applications like construction or design, this pattern often appears when working with areas of rectangular spaces. When you know one dimension of a rectangle and the total area, you can find the other dimension by dividing the area by the known dimension.

4. Liam is designing a rectangular garden with an area of (x² - 64) square meters. He knows the length is (x + 8) meters. What expression represents the width of the garden in meters? Answer: (x - 8) Solution: Area = length × width Area = x² - 64 Length = x + 8 Let width = W.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width Given: Area = x² - 64 Length = x + 8 Let width = W. So: x² - 64 = (x + 8) × W To find W, divide both sides by (x + 8): W = (x² - 64) / (x + 8) Now, notice that x² - 64 is a difference of squares: x² - 64 = x² - 8² = (x - 8)(x + 8) So: W = (x - 8)(x + 8) / (x + 8) Cancel the common factor (x + 8) (as long as x ≠ -8, which is fine for dimensions): W = x - 8 Thus, the width is (x - 8) meters.

5. 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).

6. 16x² - 81 = ? Answer: (4x - 9)(4x + 9) Solution: Step 1: Identify the pattern a² - b² where both terms are perfect squares Step 2: 16x² = (4x)² and 81 = 9² Step 3: Apply the difference of squares formula: a² - b² = (a - b)(a + b) Step 4: Substitute a = 4x and b = 9: (4x - 9)(4x + 9) Step 5: Verify by expanding: (4x - 9)(4x + 9) = 16x² + 36x -…
   Full step-by-step solution

   Step 1: Identify the pattern a² - b² where both terms are perfect squares Step 2: 16x² = (4x)² and 81 = 9² Step 3: Apply the difference of squares formula: a² - b² = (a - b)(a + b) Step 4: Substitute a = 4x and b = 9: (4x - 9)(4x + 9) Step 5: Verify by expanding: (4x - 9)(4x + 9) = 16x² + 36x - 36x - 81 = 16x² - 81 The factored form is (4x - 9)(4x + 9).

7. Tane is designing a square garden plot for a community project. The entire plot is a large square with side length (5x + 3y) meters. Inside this plot, a smaller square fountain area is planned with side length (5x - 3y) meters. The area between the large square and the small square will be filled with decorative gravel. Using the difference of squares, what is the area of the gravel region in square meters, expressed as a simplified polynomial? Answer: 60xy Solution: The area of the large square is (5x + 3y)^2. The area of the small square is (5x - 3y)^2. The area of the gravel region is (5x + 3y)^2 - (5x - 3y)^2.
   Full step-by-step solution

   Step 1: The area of the large square is (5x + 3y)^2. The area of the small square is (5x - 3y)^2. Step 2: The area of the gravel region is (5x + 3y)^2 - (5x - 3y)^2. Step 3: Recognize this as a difference of squares: a^2 - b^2 = (a + b)(a - b), where a = (5x + 3y) and b = (5x - 3y). Step 4: Calculate a + b = (5x + 3y) + (5x - 3y) = 10x. Step 5: Calculate a - b = (5x + 3y) - (5x - 3y) = 5x + 3y - 5x + 3y = 6y. Step 6: Multiply: (10x)(6y) = 60xy. Step 7: The area of the gravel region is 60xy square meters. The answer is 60xy.