Question 1: Solve for x: x^2 + 4x + 1 = 0 using completing the square.

Answer: x = -2 ± √3

Question 2: Complete the square for the expression: x^2 + 6x.

Answer: x^2 + 6x = (x + 3)^2 - 9

Question 3: Find the value of x: 2x^2 + 8x + 6 = 0 by completing the square.

Answer: x = -2 ± 1

Question 4: Construct the quadratic expression by completing the square: x^2 + 10x + 25.

Answer: x^2 + 10x + 25 = (x + 5)^2

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

Answer: x = 3 ± √4 = 3 ± 2

Question 6: A quadratic equation is given as x^2 + 8x + k = 0. If the solution by completing the square results in x = -4 ± √(k+16), find the value of k.

Answer: k = 0

Question 7: Solve: x^2 + 2x - 3 = 0 by completing the square.

Answer: x = -1 ± 2

Question 8: Explain the steps to solve the quadratic equation x^2 + 12x + 36 = 0 by completing the square.

Answer: First, rewrite as x^2 + 12x = -36. Then, add (12/2)^2 = 36 to both sides: x^2 + 12x + 36 = 0. Recognize the perfect square: (x + 6)^2 = 0. Hence, x = -6.

Question 9: In the equation x^2 + 14x + 49 = 0, identify the completed square form and find the root.

Answer: The form is (x + 7)^2 = 0, root is x = -7

Question 10: Identify and correct the mistake: To solve x^2 + 4x + 3 = 0 by completing the square, a student writes: x^2 + 4x = -3, then (x + 2)^2 = 1, and concludes x = -2 ± 1. Is this correct? If not, correct the error.

Answer: The mistake is in completing the square: (x + 2)^2 = x^2 + 4x + 4, not 1. Correctly, x^2 + 4x + 4 = 0 - 4 + 3, so (x + 2)^2 = -1, which has no real solutions.