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Special Products · Grade 9 · Worksheet 2

1. (6x + 8y)(6x - 8y) = ? Answer: 36x² - 64y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 6x and b = 8y Square the first term: (6x)² = 36x² Square the second term: (8y)² = 64y² Apply the formula: a² - b² = 36x² - 64y² The answer is 36x² - 64y².
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 6x and b = 8y Step 3: Square the first term: (6x)² = 36x² Step 4: Square the second term: (8y)² = 64y² Step 5: Apply the formula: a² - b² = 36x² - 64y² The answer is 36x² - 64y².

2. Emma is constructing a decorative rectangular frame for a large abstract painting. The length of the frame is (7x + 9) centimeters and the width is (7x - 9) centimeters. She needs to calculate the area of the frame's opening (the space inside the frame) to ensure the painting fits perfectly. What is the simplified expression for the area of the frame's opening in square centimeters? Answer: 49x² - 81 Solution: The area of the frame's opening is length × width. Area = (7x + 9)(7x - 9) This matches the difference of squares pattern: (a + b)(a - b) = a² - b². Here, a = 7x and b = 9.
   Full step-by-step solution

   Step 1: The area of the frame's opening is length × width. Step 2: Area = (7x + 9)(7x - 9) Step 3: This matches the difference of squares pattern: (a + b)(a - b) = a² - b². Step 4: Here, a = 7x and b = 9. Step 5: Apply the formula: (7x)² - (9)² Step 6: Calculate: 49x² - 81 Step 7: The simplified expression for the area is 49x² - 81 square centimeters.

3. An architect is designing a rectangular conference room where the length is (5x + 8) feet and the width is (5x - 8) feet. To determine the appropriate ventilation system, she needs to calculate the room's area. What is the simplified expression for the area of the conference room in square feet? Answer: 25x^2 - 64 Solution: The area of a rectangle is length times width. Area = (5x + 8)(5x - 8) Recognize this follows the difference of squares pattern: (a + b)(a - b) = a^2 - b^2 Here, a = 5x and b = 8 Apply the formula: (5x)^2 - (8)^2 Calculate: 25x^2 - 64 The simplified expression for the area is 25x^2 - 64 square feet.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. Step 2: Area = (5x + 8)(5x - 8) Step 3: Recognize this follows the difference of squares pattern: (a + b)(a - b) = a^2 - b^2 Step 4: Here, a = 5x and b = 8 Step 5: Apply the formula: (5x)^2 - (8)^2 Step 6: Calculate: 25x^2 - 64 Step 7: The simplified expression for the area is 25x^2 - 64 square feet.

4. An architect is designing a rectangular conference room with dimensions (5x + 3) meters by (5x - 3) meters. She needs to calculate the area to determine the flooring requirements. What is the simplified expression for the area of the conference room in square meters? Answer: 25x^2 - 9 Solution: Area = (5x + 3)(5x - 3) Apply the difference of squares formula: (a + b)(a - b) = a² - b² Here, a = 5x and b = 3 Area = (5x)² - (3)² Area = 25x² - 9 The simplified expression for the area is 25x² - 9.
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: Area = (5x + 3)(5x - 3) Step 3: Apply the difference of squares formula: (a + b)(a - b) = a² - b² Step 4: Here, a = 5x and b = 3 Step 5: Area = (5x)² - (3)² Step 6: Area = 25x² - 9 The simplified expression for the area is 25x² - 9.

5. A rectangular garden is designed with a circular fountain at its center. The garden measures (3x + 2) meters by (3x - 2) meters, and the fountain has a radius of x meters. What is the area of the garden space surrounding the fountain? Express your answer in terms of x. Answer: 8x² - 4πx² Solution: This problem combines geometric area calculations with algebraic expressions. When finding the area of a composite shape, we often calculate the total area and subtract the unwanted portions.
   Full step-by-step solution

   This problem combines geometric area calculations with algebraic expressions. When finding the area of a composite shape, we often calculate the total area and subtract the unwanted portions. The rectangular dimensions represent a difference of squares pattern, which simplifies nicely. The area of a circle involves the constant π, and when subtracting areas with algebraic terms, we combine like terms carefully.

6. An architect is designing a rectangular conference room where the length is (5x + 8) feet and the width is (5x - 8) feet. The client wants to install custom carpeting and needs to know the exact area of the room. Using the difference of squares formula, what is the simplified expression for the area of the conference room in square feet? Answer: 25x^2 - 64 Solution: The area of a rectangle is length times width. Area = (5x + 8)(5x - 8) Apply the difference of squares formula: (a + b)(a - b) = a^2 - b^2 Here, a = 5x and b = 8 Calculate a^2: (5x)^2 = 25x^2 Calculate b^2: (8)^2 = 64 Apply the formula: 25x^2 - 64 The simplified expression for the area is 25x^2…
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. Step 2: Area = (5x + 8)(5x - 8) Step 3: Apply the difference of squares formula: (a + b)(a - b) = a^2 - b^2 Step 4: Here, a = 5x and b = 8 Step 5: Calculate a^2: (5x)^2 = 25x^2 Step 6: Calculate b^2: (8)^2 = 64 Step 7: Apply the formula: 25x^2 - 64 Step 8: The simplified expression for the area is 25x^2 - 64 square feet.

7. Olivia is planning a rectangular community garden plot. The length of the plot is (7x + 9) meters, and the width is (7x - 9) meters. She needs to calculate the area of the plot to order fertilizer. What is the simplified expression for the area of the garden in square meters? Answer: 49x² - 81 Solution: The area of a rectangle is length times width. So area = (7x + 9)(7x - 9). Recognize this fits the difference of squares pattern: (a + b)(a - b) = a² - b².
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. So area = (7x + 9)(7x - 9). Step 2: Recognize this fits the difference of squares pattern: (a + b)(a - b) = a² - b². Step 3: Here, a = 7x and b = 9. Step 4: Apply the formula: (7x)² - (9)². Step 5: Calculate (7x)² = 49x². Step 6: Calculate 9² = 81. Step 7: Subtract: 49x² - 81. The simplified expression for the area is 49x² - 81 square meters.