Factoring Quadratics with Other Leading Coefficients

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

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Algebra 1 · Unit 8 · Topic 8.3

Factoring Quadratics with Other Leading Coefficients

For ax² + bx + c, one method splits bx into terms whose coefficients add to b and multiply to ac, then factors by grouping.

Definition and explanation

For ax² + bx + c, one method splits bx into terms whose coefficients add to b and multiply to ac, then factors by grouping. Not every quadratic has rational factors.

Worked example

Factor 3x² + 11x + 6. Since 9 + 2 = 11 and 9 × 2 = 18 = 3 × 6, write 3x² + 9x + 2x + 6. Grouping gives 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3).

Common mistake

Do not force an integer factorization when no suitable integer pair exists.

Key takeaway

For ax² + bx + c, one method splits bx into terms whose coefficients add to b and multiply to ac, then factors by grouping.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.