Special Products

lessonbunny.com

Answer Key & Explanations

Special Products · Grade 9 · Worksheet 2

1. (3x + 5)(3x - 5) = ? Answer: 9x² - 25 Solution: We are given: (3x + 5)(3x - 5) Recognize the pattern. This is in the form (a + b)(a - b), which equals a² - b². Here, a = 3x and b = 5.
   Full step-by-step solution

   We are given: (3x + 5)(3x - 5) Step 1: Recognize the pattern. This is in the form (a + b)(a - b), which equals a² - b². Here, a = 3x and b = 5. Step 2: Apply the formula. (a + b)(a - b) = a² - b² So, (3x + 5)(3x - 5) = (3x)² - (5)². Step 3: Calculate each term. (3x)² = 3² * x² = 9x². (5)² = 25. Step 4: Write the final result. 9x² - 25. Thus, the answer is 9x² - 25.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (2x,0), and (0,3x). The hypotenuse has a length of 5√13 units. Using the Pythagorean theorem, determine the value of x. Answer: 5 Solution: Identify the legs of the right triangle from the coordinates: horizontal leg = 2x, vertical leg = 3x Apply the Pythagorean theorem: (2x)^2 + (3x)^2 = (5√13)^2 Simplify the equation: 4x^2 + 9x^2 = 25 * 13 Combine like terms: 13x^2 = 325 Divide both sides by 13: x^2 = 25 Take the square root of…
   Full step-by-step solution

   Step 1: Identify the legs of the right triangle from the coordinates: horizontal leg = 2x, vertical leg = 3x Step 2: Apply the Pythagorean theorem: (2x)^2 + (3x)^2 = (5√13)^2 Step 3: Simplify the equation: 4x^2 + 9x^2 = 25 * 13 Step 4: Combine like terms: 13x^2 = 325 Step 5: Divide both sides by 13: x^2 = 25 Step 6: Take the square root of both sides: x = 5 (since length must be positive) The answer is 5.

3. (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.

4. 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.

5. An architect is designing a rectangular conference room with dimensions (5x + 8) feet by (5x - 8) feet. The client wants to install sound-absorbing panels that cost $12 per square foot. What simplified expression represents the total area of the conference room in square feet? Answer: 25x^2 - 64 Solution: Area = (5x + 8)(5x - 8) Apply the difference of squares formula: (a + b)(a - b) = a² - b² Here, a = 5x and b = 8 Calculate a² = (5x)² = 25x² Calculate b² = (8)² = 64 Apply the formula: 25x² - 64 The simplified expression for the area is 25x² - 64 square feet
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: Area = (5x + 8)(5x - 8) Step 3: Apply the difference of squares formula: (a + b)(a - b) = a² - b² Step 4: Here, a = 5x and b = 8 Step 5: Calculate a² = (5x)² = 25x² Step 6: Calculate b² = (8)² = 64 Step 7: Apply the formula: 25x² - 64 Step 8: The simplified expression for the area is 25x² - 64 square feet

6. 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.

7. Aroha is designing a rectangular community garden in her neighborhood. The length of the garden is (6x + 11) meters and the width is (6x - 11) meters. She needs to determine the area of the garden to order enough soil and plants. What is the simplified expression for the area of the garden in square meters? Answer: 36x² - 121 Solution: The area of a rectangle is length × width. So area = (6x + 11)(6x - 11). Recognize this as the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 6x and b = 11.
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width. So area = (6x + 11)(6x - 11). Step 2: Recognize this as the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 6x and b = 11. Step 3: Apply the formula: (6x)² - (11)². Step 4: Calculate (6x)² = 36x² and (11)² = 121. Step 5: The simplified expression is 36x² - 121. Final answer: 36x² - 121 square meters.

8. (4x³ + 3y²)(4x³ - 3y²) = ? Answer: 16x⁶ - 9y⁴ Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 4x³ and b = 3y² Square the first term: (4x³)² = 16x⁶ Square the second term: (3y²)² = 9y⁴ Apply the difference of squares formula: 16x⁶ - 9y⁴ The answer is 16x⁶ - 9y⁴.
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 4x³ and b = 3y² Step 3: Square the first term: (4x³)² = 16x⁶ Step 4: Square the second term: (3y²)² = 9y⁴ Step 5: Apply the difference of squares formula: 16x⁶ - 9y⁴ The answer is 16x⁶ - 9y⁴.