Question 1: Factorize the expression 25x^2 - 36.

Answer: (5x + 6)(5x - 6)

Question 2: Expand and simplify the expression (3x + 4)(3x - 4).

Answer: 9x^2 - 16

Question 3: Identify whether the expression 64a^2 - 81b^2 is a difference of two squares.

Answer: Yes

Question 4: Factorize 49x^2 - 16.

Answer: (7x + 4)(7x - 4)

Question 1: A rectangular garden has a length of 10m and is expanded on both sides by x meters to form a new shape. The area difference between the original and expanded garden is given by a difference of two squares. Write an expression for this difference and factorize it.

Answer: Original area: 10 * width; expanded area: (10 + 2x) * width; difference: (10 + 2x)^2 - 10^2 = (2x + 10)^2 - 10^2 = (2x + 10 + 10)(2x + 10 - 10) = (2x + 20)(2x).

Question 2: A ball of radius r is painted on its surface. The surface area difference when the radius increases by x units is an algebraic expression. Show this as a difference of squares and factorize.

Answer: Surface area: 4πr^2; new: 4π(r + x)^2; difference: 4π[(r + x)^2 - r^2] = 4π[(r^2 + 2rx + x^2) - r^2] = 4π(2rx + x^2) = 2πx(4r + x).

Question 3: A square and a rectangle share a common property related to difference of squares. If the side of the square is s and the rectangle's length is l and width w, express the difference of their areas and identify when it simplifies as a difference of two squares.

Answer: Difference of areas: s^2 - lw; for it to be a difference of squares, s^2 and lw must be perfect squares or equal to (a + b)^2 and (a - b)^2 for some a, b.

Question 1: A manufacturer produces square tiles of side (a + b). If the total area of two such tiles is 2(a^2 + b^2), show how this relates to the difference of two squares and interpret the result.

Answer: Total area: 2(a + b)^2 = 2(a^2 + 2ab + b^2); since total area equals 2(a^2 + b^2), this suggests 2(a^2 + 2ab + b^2) = 2(a^2 + b^2), implying 4ab = 0 or no real solution unless a or b is zero. Alternatively, express as difference of squares if possible.

Question 2: A rectangular window measures (x + y) meters in length and (x - y) meters in width. Find an expression for the total area and demonstrate it as a difference of two squares.

Answer: Area = (x + y)(x - y) = x^2 - y^2.

Question 1: Prove that the difference of two squares, a^2 - b^2, can be factored into (a + b)(a - b). Then, find values of a and b such that the product (a + b)(a - b) equals 45, with a and b as integers.

Answer: a^2 - b^2 = (a + b)(a - b). To find integer solutions for (a + b)(a - b) = 45, factors of 45: (1, 45), (3, 15), (5, 9). For each pair, solve for a and b: e.g., a + b = 9, a - b = 5 => a = 7, b = 2.

Question 2: A rectangular prism has length (x + y), width (x - y), and height z. If the surface area difference between the expanded and original prism involves a difference of squares, derive an expression for this difference and interpret its significance.

Answer: Original surface area: 2[(x + y)(x - y) + (x + y)z + (x - y)z]; expanded versions involve (x + y + Δ)^2 and similar terms, leading to difference expressed as a difference of squares involving x, y, z.

Question 1: Identify and factor the expression: 81m^2 - 25n^2.

Answer: (9m + 5n)(9m - 5n)

Question 2: The difference of the areas of two squares is 75. If one square has a side length of 15, find the side length of the other square.

Answer: Area difference: 75 = s^2 - 15^2 => s^2 - 225 = 75 => s^2 = 300 => s = √300 ≈ 17.32.

Question 1: A student factors 36x^2 - 49 as (6x + 7)(6x - 7). Is this correct? If not, correct the factorization and explain the mistake.

Answer: Incorrect. 36x^2 - 49 = (6x)^2 - 7^2 = (6x + 7)(6x - 7). The factorization is correct. No correction needed.

Question 2: A student claims that x^2 - 9 is not a difference of squares. Is this true? Correct the statement.

Answer: False. x^2 - 9 = (x + 3)(x - 3), which is a difference of squares.