Factor Difference Squares

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Factor Difference Squares · Grade 9 · Worksheet 3

1. Emma is designing a square mosaic for her art project. She starts with a large square tile of area (9x² - 16y²) square inches. She realizes this area represents the difference between the area of the original tile and a smaller square section she plans to remove. If the side length of the original tile is (3x + 4y) inches, what expression represents the side length of the smaller square section in inches? Answer: 3x - 4y Solution: Recognize that the area 9x² - 16y² is a difference of squares Write the area as (3x)² - (4y)² Apply the difference of squares formula: a² - b² = (a + b)(a - b) So 9x² - 16y² = (3x + 4y)(3x - 4y) We're told the original side length is (3x + 4y) inches Therefore, the smaller square section must…
   Full step-by-step solution

   Step 1: Recognize that the area 9x² - 16y² is a difference of squares Step 2: Write the area as (3x)² - (4y)² Step 3: Apply the difference of squares formula: a² - b² = (a + b)(a - b) Step 4: So 9x² - 16y² = (3x + 4y)(3x - 4y) Step 5: We're told the original side length is (3x + 4y) inches Step 6: Therefore, the smaller square section must have side length (3x - 4y) inches The answer is 3x - 4y.

2. 49x² - 81y² = ? Answer: (7x - 9y)(7x + 9y) Solution: The difference of squares is a factoring pattern where a² - b² = (a - b)(a + b). This works because when you multiply (a - b)(a + b), the middle terms cancel out.
   Full step-by-step solution

   The difference of squares is a factoring pattern where a² - b² = (a - b)(a + b). This works because when you multiply (a - b)(a + b), the middle terms cancel out. For example, x² - 16 factors as (x - 4)(x + 4) because x² is a perfect square and 16 is a perfect square.

3. Emma 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 - 64 square meters. She realizes this area represents the difference between the area of the entire roof space and the area where panels cannot be installed. If the length of the available space is (7x + 8) meters, what expression represents the width of the available space in terms of x? Answer: (7x - 8) Solution: The area is given as 49x² - 64, which is a difference of squares.
   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, so we can write: 49x² - 64 = (7x)² - 8² Step 3: Apply the difference of squares formula: a² - b² = (a + b)(a - b) Step 4: Substitute a = 7x and b = 8: (7x)² - 8² = (7x + 8)(7x - 8) Step 5: The length is given as (7x + 8) meters, so the width must be the other factor: (7x - 8) Step 6: Therefore, the width of the available space is (7x - 8) meters.

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. A rectangular garden has a length of (x + 7) meters and a width of (x - 7) meters. The area of the garden can be expressed as a difference of squares. What is the simplified algebraic expression for the area of the garden? Answer: x^2 - 49 Solution: Write down the formula for the area of a rectangle. Area = length × width Substitute the given expressions for length and width.
   Full step-by-step solution

   Step 1: Write down the formula for the area of a rectangle. Area = length × width Step 2: Substitute the given expressions for length and width. Length = (x + 7) meters Width = (x - 7) meters So, Area = (x + 7) × (x - 7) Step 3: Recognize the pattern. This is in the form (a + b)(a - b), which equals a^2 - b^2. Here, a = x and b = 7. Step 4: Apply the difference of squares formula. (a + b)(a - b) = a^2 - b^2 So, (x + 7)(x - 7) = x^2 - 7^2 Step 5: Simplify. 7^2 = 49 So, Area = x^2 - 49 Final Answer: x^2 - 49

6. 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.

7. 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)