Question 1: Solve for x: x^2 + 6x + 9 = 0 using the completing the square method.

Answer: x = -3

Question 2: Complete the square to solve: x^2 + 4x = 1.

Answer: x = -2 ± √2

Question 3: Construct a quadratic equation by completing the square that has solutions x = 2 and x = -1.

Answer: x^2 - 2x - 2 = 0

Question 4: Solve: 2x^2 + 8x + 6 = 0 using the completing the square method.

Answer: x = -1 ± √(2/3)

Question 5: Explain why completing the square is a useful method for solving quadratics in real-world contexts.

Answer: It allows solving equations where factoring is difficult and helps in deriving quadratic formula.

Question 6: Given the quadratic x^2 + 2x - 8 = 0, complete the square and find its solutions.

Answer: x = -1 ± √9

Question 7: Identify and correct the mistake in solving: x^2 + 10x + 25 = 0 by completing the square.

Answer: The mistake is assuming the quadratic is not a perfect square; it actually factors as (x + 5)^2 = 0.

Question 8: Solve the quadratic 3x^2 + 12x + 12 = 0 using the completing the square method. Show your steps.

Answer: x = -2 ± √(2/3)

Question 9: A ball is thrown such that its height h(t) = t^2 + 4t + 3 (meters) after t seconds. Use completing the square to find the time when the ball reaches its maximum height.

Answer: Maximum height at t = -2

Question 10: Simplify and solve for x: x^2 + 8x + 16 = 0 using completing the square.

Answer: x = -4

Question 11: Challenge: Derive a quadratic equation with solutions x = 5 and x = -3 by completing the square.

Answer: x^2 - 2x - 15 = 0