Question 1: Solve the quadratic equation x^2 + 6x + 5 = 0 using the completing the square method.

Answer: x = -1 or x = -5

Question 2: Complete the square to solve 2x^2 + 8x - 6 = 0.

Answer: x = -2 ± √(4/2)

Question 3: Find the roots of x^2 - 4x - 21 = 0 by completing the square.

Answer: x = 7 or x = -3

Question 4: Construct a quadratic equation with roots at x=3 and x=-5, then solve it using the alternative method.

Answer: x^2 + 2x - 15 = 0; roots at 3 and -5

Question 5: Determine the value of x if 3x^2 + 12x + 12 = 0, by completing the square.

Answer: x = -2 ± 2√2

Question 6: Solve 4x^2 - 4x + 1 = 0 by completing the square and explain your steps.

Answer: x = 1/2 (double root), as (2x - 1)^2 = 0

Question 7: Given the quadratic 5x^2 + 20x + 15 = 0, use the alternative method to find the solutions.

Answer: x = -2 ± √(4/5)

Question 8: A ball is thrown and its height (h) in meters after t seconds is given by h = -t^2 + 4t + 1. Find the time when the ball reaches its maximum height using completing the square.

Answer: t = 2 seconds

Question 9: Solve x^2 + 10x + 21 = 0 using the completing the square method, and verify your solutions.

Answer: x = -5 ± √4

Question 10: Identify and correct the mistake in solving x^2 + 2x - 8 = 0 by completing the square.

Answer: Incorrect step: Adding 1 instead of (x+1)^2; correct solution: (x+1)^2 - 9 = 0

Question 11: Challenge: Solve 3x^2 + 12x + 12 = 0 by completing the square, and express your answer in simplest radical form.

Answer: x = -2 ± 2√2