Question 1: Rewrite the quadratic expression x^2 + 6x + 5 in completed square form.

Answer: (x + 3)^2 - 4

Question 2: Solve for x: x^2 + 4x - 5 = 0 by completing the square.

Answer: x = 1 or x = -5

Question 3: Complete the square to solve: x^2 - 2x = 3.

Answer: x = 1 ± √2

Question 4: Plot the graph of y = x^2 + 4x + 3 on the grid.

Answer: Student plots parabola with vertex at (-2, -1)

Question 5: Construct a triangle on the grid with sides 3, 4, and 5. Using completing the square, find the length of the diagonal if placed as a right triangle.

Answer: Diagonal length = 5 units

Question 6: Explain why the quadratic x^2 + 2x + 1 can be written as (x + 1)^2.

Answer: Because completing the square shows x^2 + 2x + 1 is a perfect square trinomial, equal to (x + 1)^2.

Question 7: Solve: 2x^2 + 8x + 6 = 0 by completing the square.

Answer: x = -1 ± √(1/2)

Question 8: Identify the common mistake in completing the square for x^2 + 10x + 21, and correct it.

Answer: Mistake: Not adding (b/2)^2; Correct: x^2 + 10x + 25 = (x + 5)^2; then adjust for constant.

Question 9: Using completing the square, solve for x: 3x^2 + 12x + 12 = 0.

Answer: x = -2 ± √(1/3)

Question 10: Challenge: Derive a quadratic expression in completed square form for 4x^2 + 8x + 3.

Answer: 4(x + 1)^2 - 1

Question 11: Error Analysis: A student writes x^2 + 8x + 6 = (x + 4)^2 + 2, claiming it is correct. Is this correct? If not, correct the mistake.

Answer: Incorrect. (x + 4)^2 = x^2 + 8x + 16; so, x^2 + 8x + 6 = (x + 4)^2 - 10.