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Special Products · Grade 9 · Worksheet 3

1. An architect is designing a rectangular conference room with dimensions (5x + 8) feet by (5x - 8) feet. She needs to calculate the area to determine the flooring requirements. 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 × 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 is 25x^2 - 64 The answer…
   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) = 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 is 25x^2 - 64 The answer is 25x^2 - 64.

2. (8x + 9y)(8x - 9y) = ? Answer: 64x² - 81y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 8x and b = 9y Square the first term: (8x)² = 64x² Square the second term: (9y)² = 81y² Apply the formula: a² - b² = 64x² - 81y² The answer is 64x² - 81y².
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 8x and b = 9y Step 3: Square the first term: (8x)² = 64x² Step 4: Square the second term: (9y)² = 81y² Step 5: Apply the formula: a² - b² = 64x² - 81y² The answer is 64x² - 81y².

3. 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.

4. A rectangular garden has a length of (3x + 5) meters and a width of (3x - 5) meters. Maya wants to calculate the area of the garden using the difference of squares formula. What is the simplified expression for the garden's area in square meters? Answer: 9x^2 - 25 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 = (3x + 5) meters Width = (3x - 5) meters Area = (3x + 5) × (3x - 5) Step 3: Recognize that this is a difference of squares. The difference of squares formula is: (a + b)(a - b) = a^2 - b^2 Step 4: Identify a and b in the expression. Here, a = 3x and b = 5. Step 5: Apply the formula. (3x + 5)(3x - 5) = (3x)^2 - (5)^2 Step 6: Simplify each term. (3x)^2 = 9x^2 (5)^2 = 25 Step 7: Write the final simplified expression. Area = 9x^2 - 25 So, the area of the garden is 9x^2 - 25 square meters.

5. (12x + 7y)(12x - 7y) = ? Answer: 144x² - 49y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 12x and b = 7y Square the first term: (12x)² = 144x² Square the second term: (7y)² = 49y² Apply the formula: a² - b² = 144x² - 49y² The answer is 144x² - 49y².
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 12x and b = 7y Step 3: Square the first term: (12x)² = 144x² Step 4: Square the second term: (7y)² = 49y² Step 5: Apply the formula: a² - b² = 144x² - 49y² The answer is 144x² - 49y².

6. A rectangular solar panel has dimensions (3x + 7) meters by (3x - 7) meters. If the area of the panel is 275 square meters, what is the value of x? Answer: 6 Solution: Write the area expression using the given dimensions: Area = (3x + 7)(3x - 7) Apply the difference of squares formula: (a + b)(a - b) = a² - b² Expand: (3x + 7)(3x - 7) = (3x)² - (7)² = 9x² - 49 Set equal to the given area: 9x² - 49 = 275 Add 49 to both sides: 9x² = 324 Divide both sides by 9:…
   Full step-by-step solution

   Step 1: Write the area expression using the given dimensions: Area = (3x + 7)(3x - 7) Step 2: Apply the difference of squares formula: (a + b)(a - b) = a² - b² Step 3: Expand: (3x + 7)(3x - 7) = (3x)² - (7)² = 9x² - 49 Step 4: Set equal to the given area: 9x² - 49 = 275 Step 5: Add 49 to both sides: 9x² = 324 Step 6: Divide both sides by 9: x² = 36 Step 7: Take the square root of both sides: x = 6 (since dimensions must be positive) The answer is 6.

7. (14x³ + 11y²)(14x³ - 11y²) = ? Answer: 196x⁶ - 121y⁴ Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 14x³ and b = 11y² Square the first term: (14x³)² = 14² × (x³)² = 196 × x⁶ = 196x⁶ Square the second term: (11y²)² = 11² × (y²)² = 121 × y⁴ = 121y⁴ Apply the formula: a² - b² = 196x⁶ - 121y⁴ The answer is 196x⁶ - 121y⁴.
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 14x³ and b = 11y² Step 3: Square the first term: (14x³)² = 14² × (x³)² = 196 × x⁶ = 196x⁶ Step 4: Square the second term: (11y²)² = 11² × (y²)² = 121 × y⁴ = 121y⁴ Step 5: Apply the formula: a² - b² = 196x⁶ - 121y⁴ The answer is 196x⁶ - 121y⁴.

8. An architect is designing a rectangular conference room with dimensions (5x + 8) meters by (5x - 8) meters. The client wants to install acoustic panels that cost $15 per square meter. What simplified expression represents the total area where panels will be installed? Answer: 25x² - 64 Solution: The difference of squares pattern states that (a + b)(a - b) = a² - b². This algebraic identity is useful for quickly multiplying binomials where the terms are the same except for opposite signs between them.
   Full step-by-step solution

   The difference of squares pattern states that (a + b)(a - b) = a² - b². This algebraic identity is useful for quickly multiplying binomials where the terms are the same except for opposite signs between them. For example, (3y + 4)(3y - 4) would simplify to 9y² - 16 using this pattern.

9. (9x² - 16y⁴) = ? Answer: (3x - 4y²)(3x + 4y²) Solution: Recognize the pattern a² - b² = (a - b)(a + b) Identify a² = 9x², so a = 3x Identify b² = 16y⁴, so b = 4y² Apply the difference of squares formula: (3x - 4y²)(3x + 4y²) The answer is (3x - 4y²)(3x + 4y²).
   Full step-by-step solution

   Step 1: Recognize the pattern a² - b² = (a - b)(a + b) Step 2: Identify a² = 9x², so a = 3x Step 3: Identify b² = 16y⁴, so b = 4y² Step 4: Apply the difference of squares formula: (3x - 4y²)(3x + 4y²) The answer is (3x - 4y²)(3x + 4y²).