Question 1: Prove that (a + b)^2 = a^2 + 2ab + b^2 for any real numbers a and b.

Answer: Proved by expanding (a + b)^2 and comparing terms

Question 2: Simplify the expression using the identity: a^2 - 2ab + b^2.

Answer: (a - b)^2

Question 3: Show that 3(x + y) = 3x + 3y.

Answer: Distribute 3 across (x + y)

Question 1: Prove that (a + b)(a - b) = a^2 - b^2 using algebraic expansion.

Answer: Expand both sides and compare terms to show equality

Question 2: Given that a^2 + 2ab + b^2 = (a + b)^2, explain why this identity holds and prove it.

Answer: Expand (a + b)^2 and compare with a^2 + 2ab + b^2 to verify

Question 3: Show that the expression 4x^2 - 9 can be written as a difference of squares.

Answer: Rewrite as (2x)^2 - 3^2

Question 1: A rectangle has length (a + b) and width (a - b). Prove that its area is a^2 - b^2.

Answer: Area = (a + b)(a - b) = a^2 - b^2 by difference of squares

Question 2: A square's side length is (x + y). Show that its area can be expanded into x^2 + 2xy + y^2.

Answer: Area = (x + y)^2 = x^2 + 2xy + y^2

Question 1: Prove that (a^3 + b^3) = (a + b)(a^2 - ab + b^2).

Answer: Factor sum of cubes by expanding and showing the product

Question 2: Using algebraic identities, prove that the expression (x^2 + 2xy + y^2) = (x + y)^2 is valid for all real x and y.

Answer: Expand (x + y)^2 and compare to x^2 + 2xy + y^2

Question 1: Show that (a - b)^2 = a^2 - 2ab + b^2.

Answer: Expand (a - b)^2 = a^2 - 2ab + b^2

Question 2: A student claims that (a + b)^2 = a^2 + b^2. Identify and correct the mistake.

Answer: Incorrect; the correct expansion is a^2 + 2ab + b^2