Question 1: Factorise the quadratic x² + 5x + 6.

Answer: (x + 2)(x + 3)

Question 2: Factorise the quadratic x² - 7x + 12.

Answer: (x - 3)(x - 4)

Question 3: Express the quadratic x² + 2x - 8 as a product of its factors.

Answer: (x + 4)(x - 2)

Question 4: Calculate the value of x if x² + 3x + 2 = 0.

Answer: x = -1 or x = -2

Question 5: A quadratic has roots at x = 2 and x = -3. Write its quadratic form in x² + bx + c.

Answer: x² + x - 6

Question 6: Construct a quadratic with roots at x = -1 and x = 4.

Answer: (x + 1)(x - 4)

Question 7: A quadratic x² + bx + c factors into (x + 5)(x + 1). Find b and c.

Answer: b = 6, c = 5

Question 8: A quadratic is factorised as (x - 2)(x + 4). Expand and write it in the form x² + bx + c.

Answer: x² + 2x - 8

Question 9: Identify the mistake: The quadratic x² + 4x + 4 is incorrectly factored as (x + 2)(x + 2). Does this factorisation work? Explain why or why not.

Answer: Yes, it works because (x + 2)(x + 2) = x² + 4x + 4, which is correct.

Question 10: A quadratic equation x² + bx + c has roots at 1/2 and -3. Find its quadratic form.

Answer: x² + (5/2)x - 3/2

Question 11: Challenge: Given the quadratic x² + 6x + k, find the value of k such that the quadratic is factorisable over the integers.

Answer: k = 9