Special Products

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Special Products · Grade 9 · Worksheet 1

1. An architect is designing a rectangular garden with a circular fountain in the center. The garden's area is represented by (4x² + 20x + 25) square meters, while the fountain occupies (4x² - 9) square meters. What expression represents the area of the garden space available for planting flowers, and what is this area in simplified form? Answer: 20x + 34 Solution: - Total garden area = \( 4x^2 + 20x + 25 \) square meters - Fountain area = \( 4x^2 - 9 \) square meters - Available planting area = Total garden area − Fountain area Available area = \( (4x^2 + 20x + 25) - (4x^2 - 9) \) 4x^2 + 20x + 25 - 4x^2 + 9 - \( 4x^2 - 4x^2 = 0 \) - \( 20x \) stays as is…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have: - Total garden area = \( 4x^2 + 20x + 25 \) square meters - Fountain area = \( 4x^2 - 9 \) square meters - Available planting area = Total garden area − Fountain area --- **Step 2: Write the subtraction** Available area = \( (4x^2 + 20x + 25) - (4x^2 - 9) \) --- **Step 3: Distribute the negative sign** \[ 4x^2 + 20x + 25 - 4x^2 + 9 \] --- **Step 4: Combine like terms** - \( 4x^2 - 4x^2 = 0 \) - \( 20x \) stays as is - \( 25 + 9 = 34 \) So we have: \[ 20x + 34 \] --- **Step 5: Final simplified expression** The available area for planting flowers is \( 20x + 34 \) square meters. --- **Final answer:** 20x + 34

2. An architect is designing a rectangular conference room with dimensions (5x + 8) feet by (5x - 8) feet. The client wants to install acoustic panels that cost $12 per square foot. What simplified algebraic expression represents the total area of the conference room floor? Answer: 25x^2 - 64 Solution: The difference of squares is a special algebraic pattern where (a + b)(a - b) = a² - b². This occurs when you multiply two binomials that are identical except for the sign between their terms.
   Full step-by-step solution

   The difference of squares is a special algebraic pattern where (a + b)(a - b) = a² - b². This occurs when you multiply two binomials that are identical except for the sign between their terms. For example, (3y + 4)(3y - 4) would simplify to 9y² - 16 using this pattern.

3. (15x + 12y)(15x - 12y) = ? Answer: 225x² - 144y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 15x and b = 12y Square the first term: (15x)² = 225x² Square the second term: (12y)² = 144y² Apply the formula: a² - b² = 225x² - 144y² The answer is 225x² - 144y².
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 15x and b = 12y Step 3: Square the first term: (15x)² = 225x² Step 4: Square the second term: (12y)² = 144y² Step 5: Apply the formula: a² - b² = 225x² - 144y² The answer is 225x² - 144y².

4. (7x + 4y)(7x - 4y) = ? Answer: 49x² - 16y² Solution: The difference of squares pattern states that when you multiply two binomials where one is the sum of two terms and the other is the difference of the same two terms, the result equals the square of the first term minus the square of the second term.
   Full step-by-step solution

   The difference of squares pattern states that when you multiply two binomials where one is the sum of two terms and the other is the difference of the same two terms, the result equals the square of the first term minus the square of the second term. For example, (5m + 2n)(5m - 2n) would equal 25m² - 4n².

5. (9x³ + 11y²)(9x³ - 11y²) = ? Answer: 81x⁶ - 121y⁴ Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 9x³ and b = 11y² Square the first term: (9x³)² = 81x⁶ Square the second term: (11y²)² = 121y⁴ Apply the formula: a² - b² = 81x⁶ - 121y⁴ The answer is 81x⁶ - 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 = 9x³ and b = 11y² Step 3: Square the first term: (9x³)² = 81x⁶ Step 4: Square the second term: (11y²)² = 121y⁴ Step 5: Apply the formula: a² - b² = 81x⁶ - 121y⁴ The answer is 81x⁶ - 121y⁴.

6. Mason is designing a rectangular metal plate for a structural support beam. The length of the plate is (7x + 9) centimeters and the width is (7x - 9) centimeters. To calculate the amount of material needed, he must find the area of the plate. What is the simplified expression for the area of the metal plate in square centimeters? Answer: 49x² - 81 Solution: Area of a rectangle = length × width, so Area = (7x + 9)(7x - 9). Recognize this matches the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 7x and b = 9.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so Area = (7x + 9)(7x - 9). Step 2: Recognize this matches the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 7x and b = 9. Step 3: Apply the formula: (7x)² - (9)². Step 4: Calculate each square: (7x)² = 49x² and (9)² = 81. Step 5: Subtract: 49x² - 81. The simplified expression for the area is 49x² - 81 square centimeters.

7. An architect is designing a rectangular conference room where the length is (4x + 3) meters and the width is (4x - 3) meters. The client wants to install acoustic panels that cost $25 per square meter. What is the simplified expression for the total area of the conference room in square meters? Answer: 16x^2 - 9 Solution: Area = (4x + 3)(4x - 3) Recognize this follows the difference of squares pattern: (a + b)(a - b) = a² - b² Here, a = 4x and b = 3 Apply the formula: (4x)² - (3)² Calculate: 16x² - 9 The simplified expression for the area is 16x² - 9 square meters
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: Area = (4x + 3)(4x - 3) Step 3: Recognize this follows the difference of squares pattern: (a + b)(a - b) = a² - b² Step 4: Here, a = 4x and b = 3 Step 5: Apply the formula: (4x)² - (3)² Step 6: Calculate: 16x² - 9 Step 7: The simplified expression for the area is 16x² - 9 square meters

8. A rectangular garden has a length of (2x + 5) meters and a width of (2x - 5) meters. The owner wants to install a decorative stone border around the entire perimeter. Write a simplified expression for the area of the garden in terms of x. Answer: 4x² - 25 Solution: We are given the length and width of a rectangular garden: Length = (2x + 5) meters Width = (2x - 5) meters We need to find the area in terms of x.
   Full step-by-step solution

   Step 1: Understand the problem We are given the length and width of a rectangular garden: Length = (2x + 5) meters Width = (2x - 5) meters We need to find the area in terms of x. Step 2: Recall the area formula for a rectangle Area = Length × Width Step 3: Substitute the given expressions Area = (2x + 5) × (2x - 5) Step 4: Recognize a special product This is in the form (a + b)(a - b) where: a = 2x b = 5 We know (a + b)(a - b) = a² - b² Step 5: Apply the formula a² = (2x)² = 4x² b² = (5)² = 25 So, Area = 4x² - 25 Step 6: Final simplified expression The area of the garden is 4x² - 25 square meters. This is already simplified because it is a difference of squares and no like terms remain.