1Cademy - Factoring $$m^2 - 13mn + 12n^2$$

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Factoring Trinomials of the Form $x^2 + bxy + cy^2$

Example

Factoring $m^2 - 13mn + 12n^2$

To factor the trinomial $m^2 - 13mn + 12n^2$ , apply the strategy for two-variable trinomials. Set up two binomials that start with $m$ and end with $n$ . Since the last term is positive ( $12n^2$ ) and the middle term is negative ( $-13mn$ ), find two negative numbers that multiply to $12$ and add to $-13$ . The negative factor pairs of $12$ are $-1, -12$ ; $-2, -6$ ; and $-3, -4$ . The pair $-1$ and $-12$ adds up to $-13$ . Place $-1$ and $-12$ as the coefficients of $n$ in the binomials. The factored expression is $(m - n)(m - 12n)$ . Check by multiplying: $(m - n)(m - 12n) = m^2 - 12mn - mn + 12n^2 = m^2 - 13mn + 12n^2$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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