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Skill 1 of 12

Factor quadratic expressions

Factor quadratic expressions ax2+bx+cax^2+bx+c, including a greatest common factor, a≠1a \neq 1, difference of squares, and perfect-square trinomials.

Common Core standards
A-SSE.2A-SSE.3a
Builds on
Nothing earlier in this unit. This is where it starts.

Common mistakes

What students most often get wrong with this skill:

  • Choosing factors whose product is cc but whose sum is not bb

  • Sign errors when cc is negative

  • Forgetting to factor out the greatest common factor first

  • Treating x2+9x^2 + 9 as factorable over the reals like a difference of squares

A worked example

From the lesson, with every step explained.

Example 1: a monic trinomial

Problem: Factor x2−x−12x^2-x-12.

Plan: There is no common factor and a=1a=1. So I need two numbers with product −12-12 and sum −1-1.

1) Here b=−1b=-1 and c=−12c=-12.
2) The product is negative, so the two numbers have opposite signs.
3) List pairs with product −12-12 and look at their sums:

1⋅(−12)1\cdot(-12) has sum −11-11.

2⋅(−6)2\cdot(-6) has sum −4-4.

3⋅(−4)3\cdot(-4) has sum −1-1. This is the pair we want.

4) Use m=3m=3 and n=−4n=-4 in the rule (x+m)(x+n)(x+m)(x+n).

Answer:

x2−x−12=(x+3)(x−4)x^2-x-12=(x+3)(x-4)

Check: multiply back.

(x+3)(x−4)=x2−4x+3x−12(x+3)(x-4)=x^2-4x+3x-12
=x2−x−12=x^2-x-12

This matches the original, so the factoring is correct.

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