Question 1: Solve for x: x^2 + 6x + 5 = 0 by completing the square.

Answer: -1 or -5

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

Answer: x = 2 ± √2

Question 3: Express the quadratic x^2 + 8x + 7 in completed square form.

Answer: (x + 4)^2 - 9

Question 4: Fluency & Practice: Solve x^2 + 10x + 16 = 0 by completing the square.

Answer: x = -5 ± 3

Question 5: Problem Solving & Reasoning: A designer wants to create a parabolic arch with the equation y = x^2 + 4x + 3. Find the vertex form and determine the maximum height of the arch.

Answer: Vertex form: y = (x + 2)^2 - 1; maximum height at x = -2, y = -1

Question 6: Construct a quadratic in standard form that models a bridge arch reaching height 5 at x = 2, with the vertex at (2, 5).

Answer: y = -(x - 2)^2 + 5

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

Answer: x = -1 ± √(2/3)

Question 8: A rectangular window's dimensions are modeled by the quadratic equation 3x^2 + 12x + 4 = 0. Find the width that maximizes the window's size using completing the square.

Answer: Width = -2, maximum size at x = -2

Question 9: Challenge & Extension: The height of an architectural arch is modeled by y = -x^2 + 6x + 2. Find the maximum height and the x-coordinate where it occurs.

Answer: Maximum height: 11 at x = 3

Question 10: Error Analysis: A student attempts to solve x^2 + 4x + 3 = 0 by completing the square, but forgets to balance the equation after adding 4 inside the parentheses. Identify and correct the mistake.

Answer: Incorrect step: (x + 2)^2 = 0; Corrected: (x + 2)^2 = 1