Question 1: Complete the square to solve: x^2 + 6x = 7.

Answer: x = -3 ± √(2)

Question 2: Rewrite the quadratic x^2 + 4x + 1 = 0 by completing the square.

Answer: x + 2)^2 = 5

Question 3: Solve for x: x^2 + 8x + 16 = 0 using completing the square.

Answer: x = -4

Question 1: A ball is thrown upward and its height h(t) in meters is modeled by h(t) = -4t^2 + 16t + 3. Complete the square to find the time when the ball reaches its maximum height.

Answer: t = 2 seconds, maximum height = 19 meters

Question 2: Given the quadratic equation x^2 + 10x + 6 = 0, students made an error by completing the square incorrectly. Identify and explain the mistake.

Answer: Incorrectly added (10/2)^2 = 25 but didn't balance the equation properly.

Question 1: A rectangular garden has a length that is 4 meters more than its width. Its area is modeled by A(w) = w^2 + 4w. Complete the square to find the width when the area is 36 square meters.

Answer: w = 3 or -7 (discard negative), so width = 3 meters

Question 1: Solve the quadratic: 2x^2 - 8x + 5 = 0 by completing the square, showing all steps.

Answer: x = 1 ± (√3)/2

Question 2: Explain why completing the square might be more advantageous than the quadratic formula in certain situations.

Answer: It provides a visual understanding and is straightforward when the quadratic is easily factorable.

Question 1: A student attempted to solve x^2 + 6x = 9 by completing the square but made an error. The student's steps are: (x + 3)^2 = 0. What mistake did they make?

Answer: They forgot to subtract 9 from both sides before completing the square.

Question 2: Identify the mistake: students tried to complete the square for x^2 + 2x + 1 = 0 and wrote (x + 1)^2 = 0 without solving for x.

Answer: They didn't take the square root to find the solutions; the solutions are x = -1.