Question 1: Simplify the expression: 49 - 25 using the difference of two squares.

Answer: 24

Question 2: Factor the algebraic expression: x^2 - 81.

Answer: (x - 9)(x + 9)

Question 3: Identify the pattern in the sequence: 36, 25, 16, 9, ... and provide the next term.

Answer: 4

Question 4: Express x^2 - 36 as a product of two binomials.

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

Question 5: The difference between two numbers is 48. If the difference of their squares is 192, find the numbers.

Answer: 8 and 16

Question 6: Observe the pattern: (x)^2 - (x-2)^2. Simplify and explain what pattern emerges as x increases.

Answer: 4x - 4, which shows the difference increases linearly with x

Question 7: Construct a quadratic expression that factors as a difference of two squares and involves a coefficient outside the squared terms.

Answer: 4x^2 - 25 = (2x - 5)(2x + 5)

Question 8: Identify the pattern: 81, 64, 49, 36, ... and write the next two terms.

Answer: 25, 16

Question 9: A rectangle's length is x + 3 and width is x - 3. Write an expression for its area and factor it if possible.

Answer: x^2 - 9, which factors as (x - 3)(x + 3)

Question 10: Find two numbers whose squares' difference is 0, but the numbers are not equal. Explain why this is impossible.

Answer: Impossible, because if their squares are equal, then the numbers are equal (by the property of squares).

Question 11: Identify the common mistake in expanding (x + 5)(x - 5) and correct it.

Answer: Incorrect: x^2 - 25. Correct expansion: x^2 - 5x + 5x - 25 = x^2 - 25 (which is correct). The mistake is forgetting to include the middle terms; the pattern is difference of squares, which is correct here, but students often forget the middle terms in other expansions.