Advanced Factoring
Factoring Quadratics with a Leading Coefficient
When the coefficient of x² is not 1, both the first and last terms influence the factor pair. One reliable approach is to multiply a × c and split the middle term.
What changes when a is not 1?
In ax² + bx + c, the first terms in the brackets must multiply to ax² while the last terms multiply to c. The cross-products must combine to bx.
Use the a × c method
- Multiply a by c.
- Find two numbers that multiply to ac and add to b.
- Split bx using those numbers.
- Factor the resulting terms by grouping.
- Expand to verify.
Worked example
Factor 2x² + 7x + 3
- a × c = 2 × 3 = 6
- 1 × 6 = 6 and 1 + 6 = 7
- 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1)
2x² + 7x + 3 = (2x + 1)(x + 3).
Common mistakes
- Looking only for factors of c.
- Splitting the middle term with numbers that do not add to b.
- Stopping before factoring by grouping.
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