Question 1: Prove that (a + b)^2 = a^2 + 2ab + b^2 for algebraic variables a and b.

Answer: a^2 + 2ab + b^2

Question 2: Show that the difference of squares: a^2 - b^2 = (a + b)(a - b).

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

Question 3: Verify the identity: 3(x + y) = 3x + 3y.

Answer: 3x + 3y

Question 4: Prove that (x - y)^2 = x^2 - 2xy + y^2.

Answer: x^2 - 2xy + y^2

Question 5: Using the identity (a + b)^2 = a^2 + 2ab + b^2, expand (2x + 3)^2.

Answer: 4x^2 + 12x + 9

Question 6: Show that 4(a + b)^2 = 4a^2 + 8ab + 4b^2.

Answer: 4a^2 + 8ab + 4b^2

Question 7: Construct a triangle where the sides satisfy the identity (a + b)^2 = a^2 + 2ab + b^2. Describe the sides using algebraic expressions.

Answer: A triangle with sides of lengths a and b, where the square of the sum of the sides equals a^2 + 2ab + b^2.

Question 8: Prove that (x + y)^2 = x^2 + 2xy + y^2 by expanding and simplifying both sides.

Answer: Expanding (x + y)^2 gives x^2 + 2xy + y^2, confirming the identity.

Question 9: Prove the identity: (a - b)^2 = a^2 - 2ab + b^2. Then, verify with specific values a=5, b=3.

Answer: Expanding gives a^2 - 2ab + b^2; substituting values yields 25 - 30 + 9 = 4.

Question 10: Identify and correct the mistake in the following proof: (a + b)^2 = a^2 + b^2.

Answer: The correct expansion is a^2 + 2ab + b^2; the mistake is missing the 2ab term.

Question 11: Simplify and prove: 2(a + b) = 2a + 2b using algebraic identities.

Answer: 2a + 2b