Quick answer: To factor ax² + bx + c, find two numbers with product ac and sum b. Split bx using those numbers, then group. Not every quadratic factors over the integers.
Worked example
Factor 6x² + 11x + 3.
- Compute ac = 18. The numbers 9 and 2 have product 18 and sum 11.
- Write 6x² + 9x + 2x + 3.
- Group: 3x(2x + 3) + 1(2x + 3).
- Factor out the common bracket: (3x + 1)(2x + 3).
Check the result
Expanding gives 6x² + 9x + 2x + 3, exactly the original polynomial.
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Common mistake to avoid
For a ≠ 1, using product c instead of ac can give the wrong split. Check both the sum and product before grouping.
Try it yourself
1. Factor x² − 5x + 6. 2. Factor 2x² − 7x + 3.
Answers and reasoning
1. Numbers −2 and −3 give (x − 2)(x − 3). 2. Split −7x as −6x − x to get (2x − 1)(x − 3).
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