Question 1: Factorise the quadratic: x² + 5x + 6 = 0 and find the solutions for x.

Answer: x = -2, x = -3

Question 2: Factorise and solve: x² - 3x - 10 = 0.

Answer: x = 5, x = -2

Question 3: Construct the quadratic equation that factors as (x + 4)(x - 1) = 0.

Answer: x² + 3x - 4 = 0

Question 4: Solve for x: x² + 2x - 8 = 0.

Answer: x = 2, x = -4

Question 5: A rectangle's length and width are such that the area is modeled by x² + 7x + 10, where x is the length. Find the dimensions if the area is 0.

Answer: x = -2, x = -5

Question 6: Factorise: 2x² + 8x = 0.

Answer: x = 0, x = -4

Question 7: Given the quadratic x² + bx + c = 0 with solutions x = -1 and x = 4, find b and c.

Answer: b = -3, c = -4

Question 8: Determine whether the quadratic x² + 2x + 1 = 0 can be factorised and find its solutions.

Answer: Yes, (x + 1)² = 0, x = -1

Question 9: Identify the error: Students attempted to solve x² + 4x + 5 = 0 by factoring into (x + 2)(x + 2) = 0. What is wrong?

Answer: They incorrectly assumed the quadratic could be factored as perfect square, but 5 is not a perfect square and the quadratic does not factor into (x + 2)².

Question 10: Solve the quadratic: x² + 6x + 9 = 0, using factorising.

Answer: x = -3

Question 11: Challenge: The quadratic equation x² + px + q = 0 has roots that sum to -5 and multiply to 6. Find p and q, then solve the quadratic.

Answer: p = 5, q = 6; solutions: x = -2, x = -3