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

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

Question 2: Identify the mistake in the following factorisation: x² + 7x + 10 = (x + 5)(x + 2).

Answer: Incorrect factorisation; the correct factors are (x + 5)(x + 2), but the mistake was assuming the middle term was 5x instead of 7x.

Question 3: Factorise x² - 9x + 20. Check your result by expanding.

Answer: (x - 4)(x - 5), check: x² - 5x - 4x + 20 = x² - 9x +20

Question 4: A student incorrectly factors x² + 3x + 2 as (x + 1)(x + 2). Explain the mistake and provide the correct factorisation.

Answer: The mistake is that (x + 1)(x + 2) expands to x² + 3x + 2, which is correct. So, in this case, no mistake. But if they had said (x + 1)(x + 1), that would be incorrect.

Question 5: Factorise x² + 2x - 8. Show all steps.

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

Question 6: A quadratic x² + bx + c is given as x² - 4x + 4. Factorise and explain why it is a perfect square.

Answer: Factorisation: (x - 2)². Because c = (b/2)², indicating a perfect square.

Question 7: The quadratic x² + 6x + 9 is factored as (x + 3)(x + 3). Is this correct? Justify your answer.

Answer: Yes, because (x + 3)(x + 3) expands to x² + 6x + 9, confirming it's a perfect square.

Question 8: Factorise x² + bx + c, where b = -2 and c = 1. What are the values of b and c? Show your solution.

Answer: x² - 2x + 1 = (x - 1)²; b = -2, c = 1

Question 9: Identify and correct the error in the factorisation: x² + 8x + 15 = (x + 3)(x + 5).

Answer: Incorrect. The correct factorisation is (x + 3)(x + 5). The mistake is in the initial statement, which is actually correct; students should see that the initial factorisation is valid, so no correction needed.

Question 10: Construct a quadratic of the form x² + bx + c that has roots at x = 2 and x = -3. Write the quadratic and factorise it.

Answer: Quadratic: (x - 2)(x + 3) = x² + x - 6

Question 11: Choose the correct factorisation for x² + 4x + 4. Is it (x + 2)(x + 2) or (x + 4)(x + 1)? Justify your choice.

Answer: (x + 2)(x + 2) because it expands to x² + 4x + 4.