Example: Factor $x^2 - 6x - 16$

1. Identify numbers that multiply to give -16 and add to give -6:

• Start by listing all factors of $16$ in pairs, ignoring the sign:
• $1, 16$
• $2, 8$
• $4, 4$
• See which of these pairs add to make -6. The fact that there are negatives in the quadratic means we use negatives here.
• 2 - 8 = -6
• Therefore, 2 and -8 are the numbers.

2. Write the quadratic in factored form:

• x^2 - 6x - 16 = (x + 2)(x - 8)

Example: Factor $4x^2 + 8x + 3$ Identify potential brackets:

• Possible factors could be:
• $(2x)(2x)$
• $(4x)(x)$

3. Multiply the first and last numbers together:

• 4 × 3 = 12 Find two numbers that multiply to make this number ($12$) and add to make the middle number ($8$):

• In this example, numbers must multiply to make $12$ and add to make $8$.

4. Find two numbers that meet the above criteria:

• 6 and 2
• $6 \times 2 = 12$, $6 + 2 = 8$

5. Write the quadratic out again but this time split the $x$ term into two parts using the pair of numbers you have just found:

• $4x^2 + 8x + 3$
• $4x^2 + 6x + 2x + 3$

6. Fully factorize each pair of terms:

• $2x(2x + 3) + 1(2x + 3)$
• Notice the brackets contain the same expression. This reassures us we are right.

7. Factorize again:

• (2x + 3)(2x + 1)

Example: Factor $6p^2 + 5p - 1$ 8. Identify two numbers that multiply to -$6$ and add to $5$:

• (6, -1)

9. Write the quadratic out, splitting the middle term:

• $6p^2 + 6p - p - 1$

10. Factorize each pair of terms:

• $6p(p + 1) - 1(p + 1)$

11. Factorize again:

• (p + 1)(6p - 1)

Example: Factor $8x^2 + 19x + 6$ 12. Identify two numbers that multiply to 48 $(8 \times 6)$ and add to $19$:

• Possible pairs:
• $(1, 48)$
• $(2, 24)$
• (3, 16)

13. Write the quadratic out, splitting the middle term:

• $8x^2 + 3x + 16x + 6$

14. Factorize each pair of terms:

• $x(8x + 3) + 2(8x + 3)$

15. Factorize again:

• (8x + 3)(x + 2)

Example: Solve $6 + 23x - 4x^2 = 0$ 16. Rewrite the equation:

• $-4x^2 + 23x + 6 = 0$

17. Factorize the quadratic:

• Factor pairs for -4 × 6 = -24 that add up to $23$:
• (24, -1)

18. Write the quadratic with the middle term split:

• $-4x^2 + 24x - x + 6 = 0$

19. Factorize by grouping:

• $4x(x - 6) - 1(x - 6) = 0$

20. Factor out the common term:

• $(4x - 1)(x - 6) = 0$

21. Set each factor to zero and solve:

• $4x - 1 = 0$

• $4x = -1$

• x = -1/4

• $x - 6 = 0$

• x = 6 Two things multiply together to make zero, so one (or both) must be zero:

• x = 6

• x = -1/4

In timed conditions, you always want to save time where possible. However, don't forget the method marks that you get by solving it manually - use your calculator to double check your answers!