Special Products

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Special Products · Grade 9 · Worksheet 1

1. 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.

2. (7x + 2y)(7x - 2y) = ? Answer: 49x² - 4y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 7x and b = 2y Square the first term: (7x)² = 49x² Square the second term: (2y)² = 4y² Apply the formula: a² - b² = 49x² - 4y² The answer is 49x² - 4y².
   Full step-by-step solution

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

3. A rectangular garden has a length of (3x + 5) meters and a width of (3x - 5) meters. The owner wants to install a stone pathway around the perimeter that requires knowing the exact area of the garden bed. What is the area of the garden in simplified form? Answer: 9x² - 25 Solution: Identify the formula for the area of a rectangle. Area = length × width Write down the given expressions for length and width. Length = (3x + 5) meters Width = (3x - 5) meters Set up the multiplication for the area.
   Full step-by-step solution

   Step 1: Identify the formula for the area of a rectangle. The area of a rectangle is given by: Area = length × width Step 2: Write down the given expressions for length and width. Length = (3x + 5) meters Width = (3x - 5) meters Step 3: Set up the multiplication for the area. Area = (3x + 5) × (3x - 5) Step 4: Recognize the special product pattern. This is in the form (a + b)(a - b), which equals a² - b². Here, a = 3x and b = 5. Step 5: Apply the formula (a + b)(a - b) = a² - b². a² = (3x)² = 9x² b² = (5)² = 25 Step 6: Subtract the squares. Area = 9x² - 25 Step 7: State the simplified area. The area of the garden in simplified form is 9x² - 25 square meters.

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. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (3x - 5) meters. He needs to calculate the area of the garden to determine how much soil to purchase. What is the area of Liam's garden in simplified form? Answer: 9x² - 25 Solution: To find the area of a rectangle, we multiply its length by its width. Write down the expressions for length and width. Length = (3x + 5) meters Width = (3x - 5) meters Write the formula for area.
   Full step-by-step solution

   To find the area of a rectangle, we multiply its length by its width. Step 1: Write down the expressions for length and width. Length = (3x + 5) meters Width = (3x - 5) meters Step 2: Write the formula for area. Area = Length × Width Area = (3x + 5) × (3x - 5) Step 3: Recognize the pattern. Notice that we are multiplying two binomials of the form (a + b)(a - b). This is a special product called the difference of squares. The rule is: (a + b)(a - b) = a² - b². Step 4: Identify 'a' and 'b'. In our expression (3x + 5)(3x - 5): a = 3x b = 5 Step 5: Apply the difference of squares formula. Area = (a)² - (b)² Area = (3x)² - (5)² Step 6: Simplify the expression. (3x)² means (3x) × (3x). We square the coefficient and the variable: 3² × x² = 9 × x² = 9x². (5)² = 25. So, Area = 9x² - 25 Therefore, the area of Liam's garden in simplified form is 9x² - 25 square meters.

6. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (3x - 5) meters. He needs to calculate the area of the garden to determine how much soil to purchase. 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 So, Area = (3x + 5) × (3x - 5) Step 3: Recognize the difference of squares pattern. The difference of squares formula is: (a + b)(a - b) = a² - b² Step 4: Identify a and b in the expression. Here, a = 3x and b = 5. Step 5: Apply the difference of squares formula. (3x + 5)(3x - 5) = (3x)² - (5)² Step 6: Simplify the squares. (3x)² = 9x² (5)² = 25 Step 7: Write the final simplified expression. Area = 9x² - 25 This is the simplified expression for the garden's area in square meters.

7. 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.

8. An architect is designing a rectangular conference room where the length is (5x + 8) feet and the width is (5x - 8) feet. To calculate the flooring cost, she needs to determine the area of the room. What is the simplified expression for the area of the conference room in square feet? Answer: 25x^2 - 64 Solution: The difference of squares is a pattern in algebra where (a + b)(a - b) = a² - b². This occurs because when you multiply the binomials, the middle terms cancel each other out.
   Full step-by-step solution

   The difference of squares is a pattern in algebra where (a + b)(a - b) = a² - b². This occurs because when you multiply the binomials, the middle terms cancel each other out. For example, (2y + 3)(2y - 3) would simplify to 4y² - 9.