Question 1: Factor the expression: a² - b².

Answer: (a - b)(a + b)

Question 2: Simplify the difference of squares: 49 - 64.

Answer: -15

Question 3: Express as a product: x² - 36.

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

Question 4: Calculate: 81 - 25.

Answer: 56

Question 1: Factor completely: 16a² - 25b².

Answer: (4a - 5b)(4a + 5b)

Question 2: Explain why the expression x² - 49 can be factored as (x - 7)(x + 7).

Answer: Because it is a difference of squares, which factors into (x - 7)(x + 7).

Question 3: A rectangle has sides of length a and b. If a² - b² = 0, what can you say about the sides?

Answer: They are equal in length, since a² - b² = (a - b)(a + b) = 0 implies a = b.

Question 4: Solve for x: x² - 16 = 0.

Answer: x = ±4

Question 5: Factor the expression: 36x² - 49.

Answer: (6x - 7)(6x + 7)

Question 1: A garden is rectangular with length a and width b. The difference of their squares is 45. What does this tell you about the relationship between a and b?

Answer: It means (a - b)(a + b) = 45, so either a = b or the sides relate such that their sum and difference multiply to 45.

Question 2: In a city, the difference between the squares of two given distances is 64. If one distance is 8 units, what is the other?

Answer: The other distance could be 16 or 0 (since 8² - 0²=64 and 16² - 8²=64).

Question 1: Prove that the difference of two squares can always be factored over the integers if both squares are perfect squares.

Answer: Because (a² - b²) = (a - b)(a + b), and if a² and b² are perfect squares, then a and b are integers, making the factorization valid over the integers.

Question 2: Given that x² - y² = 77, list all possible integer pairs (x, y).

Answer: Pairs where (x - y)(x + y) = 77; factors of 77 are (1, 77) and (7, 11). Possible pairs: (x, y) = (39, 38), (39, -38), (-39, 38), (-39, -38).

Question 1: Simplify: a² - 36 as (a - 6)(a + 6). Is this correct? Explain.

Answer: Yes, it is correct because a² - 36 is a difference of squares: a² - 6² = (a - 6)(a + 6).

Question 2: Identify the error: Factor x² + 9 as (x + 3)(x - 3).

Answer: Incorrect, because x² + 9 cannot be factored as a difference of squares; it is a sum, which does not factor over the real numbers.

Question 3: Factor: 100a² - 25.

Answer: (10a - 5)(10a + 5) or 25(4a² - 1) which further factors to 25(2a - 1)(2a + 1).