Question 1: Solve the quadratic equation x^2 + 6x + 5 = 0 by completing the square using the alternative method.

Answer: x = -1 or x = -5

Question 2: Find the value of x in the equation 4x^2 - 12x + 9 = 0 using the alternative completing the square method.

Answer: x = 3/2

Question 3: Determine the roots of x^2 + 8x + 16 = 0 through the alternative method, explaining each step.

Answer: Completing the square: x^2 + 8x + 16 = (x + 4)^2 = 0; so, x = -4.

Question 4: Construct a quadratic equation with roots 2 and -3, then solve it using the alternative method.

Answer: Equation: (x - 2)(x + 3) = 0 => x^2 + x - 6 = 0. Completing the square: x^2 + x = 6; (x + 0.5)^2 = 6 + 0.25 = 6.25; x + 0.5 = ±√6.25 = ±2.5; x = -0.5 ± 2.5

Question 5: Solve x^2 + 10x + 25 = 0 using the alternative completing the square method.

Answer: x = -5

Question 6: A ball is thrown upwards so that its height in meters after t seconds is given by h(t) = t^2 + 4t + 3. Find the time when it reaches its maximum height using the alternative method.

Answer: Completing the square: t^2 + 4t + 3 = (t + 2)^2 - 4 + 3 = (t + 2)^2 - 1; maximum at vertex: t = -b/2a = -4/2 = -2 seconds (since parabola opens upwards, this is the minimum height, so for maximum, check the vertex; here, as it opens upwards, the maximum height is at the vertex, which is t = -2).

Question 7: Solve 3x^2 - 12x + 12 = 0 by completing the square using the alternative method.

Answer: x = 2 ± √2

Question 8: Using the alternative method, solve x^2 + 2x - 8 = 0.

Answer: x = -1 ± √9 = -1 ± 3; so x = 2 or x = -4

Question 9: Identify and correct the mistake in solving x^2 + 6x + 9 = 0 by completing the square if a student incorrectly obtains x = -3.

Answer: Mistake: They treated the equation as (x + 3)^2 = 0, which is correct; x = -3 is correct. But if they found x = 3, they made an error. The correction is recognizing that (x + 3)^2 = 0 implies x = -3.

Question 10: Challenge: Find the roots of 2x^2 + 8x + 6 = 0 using the alternative method and express your answer in simplest radical form.

Answer: Divide through by 2: x^2 + 4x + 3 = 0; Complete the square: (x + 2)^2 - 4 + 3 = 0; (x + 2)^2 = 1; x + 2 = ±1; x = -2 ± 1 => x = -1 or x = -3