Question 1: Rewrite the quadratic equation x^2 + 6x + 5 = 0 in completed square form.

Answer: (x + 3)^2 - 4 = 0

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

Answer: x = -2 ± √2

Question 3: Construct and complete the square for the quadratic expression x^2 + 4x + 7.

Answer: (x + 2)^2 + 3

Question 4: Identify and correct the mistake in the following solution: The student writes x^2 + 4x + 3 = (x + 2)^2 - 1 and then solves for x.

Answer: Incorrect completing the square; should be (x + 2)^2 - 1, but original was x^2 + 4x + 3, which completes to (x + 2)^2 - 1. The mistake is in the initial expansion or the sign, or the student incorrectly simplified.

Question 1: Express x^2 + 10x + 16 in completed square form.

Answer: (x + 5)^2 - 9

Question 2: Solve x^2 + 2x - 3 = 0 by completing the square.

Answer: x = -1 ± √2

Question 3: Complete the square for the quadratic x^2 - 6x + 9.

Answer: (x - 3)^2

Question 1: A ball is launched such that its height h (in meters) after t seconds is given by h = -4t^2 + 12t + 2. Use completing the square to find the maximum height and the time at which it occurs.

Answer: Maximum height is 8 meters at t = 1.5 seconds.

Question 2: A quadratic equation models the profit P (in thousands) for a product: P = -3x^2 + 12x - 4. Find the value of x that maximizes profit, using completing the square.

Answer: x = 2

Question 1: A rectangular garden has a length of (x + 3) meters and a width of (x - 1) meters. If the total area is 60 square meters, find the value of x.

Answer: x = 4

Question 1: Solve the quadratic equation 3x^2 - 12x + 7 = 0 by completing the square and express your answer in simplest form.

Answer: x = 2 ± √(5/3)

Question 2: Prove that the quadratic equation x^2 + 8x + 16 = 0 has a perfect square form, and find its root(s).

Answer: (x + 4)^2 = 0, root x = -4