0580
Expanding and Factorising Brackets
Algebra and Sequences
- For ax² + bx + c, find a pair multiplying to ac and adding to b
- Rewrite bx as the sum of those two terms, which turns three terms into four
- A common bracket appears, and taking it out completes the factorisation
Worked example
Factorising a harder quadratic
Factorise 3x² + 11x + 6.
Solution:
- A number in front of x² means the two numbers no longer simply multiply to the constant, so use the grouping method
- Write the expression as ax² + bx + c, so a = 3, b = 11 and c = 6
- Find two numbers that multiply to ac and add to b
- ac = 3 × 6 = 18, and b = 11
- Factor pairs of 18: 1 × 18, 2 × 9, 3 × 6. The pair 9 and 2 adds to 11
- Split the middle term using those two numbers: 3x² + 9x + 2x + 6
- Factorise the first pair and the second pair separately: 3x(x + 3) + 2(x + 3)
- Both now carry the bracket (x + 3), which confirms the split was right — if the brackets differ, the pair was wrong
- Take the common bracket out: (x + 3)(3x + 2)
- Check by expanding: 3x² + 2x + 9x + 6 = 3x² + 11x + 6