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

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

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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 2026-06-03

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OpenStax Intermediate Algebra

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As a logistics coordinator, you use a cost-estimation model where the resource allocation is represented by the expression $m^2 - 13mn + 12n^2$ (where $m$ represents machine hours and $n$ represents the number of network nodes). To properly enter this into the company's legacy tracking database, you must recall and input its factored equivalent. Which of the following represents the correct factored form of this expression?
In a logistics cost-estimation model, the expression $m^2 - 13mn + 12n^2$ represents the relationship between machine hours ( $m$ ) and network nodes ( $n$ ). When simplifying this model for a budget report, the expression is factored into the form $(m - n)(m - kn)$ . Based on the factoring rules for two-variable trinomials, what is the value of the missing numerical coefficient $k$ ?
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In a resource allocation analysis, a team uses the expression $m^2 - 13mn + 12n^2$ to model variable maintenance costs. To simplify this model for a budget report, you must identify the specific numerical components involved in its factorization. Match each factoring requirement or component with its correct numerical value based on the standard strategy for two-variable trinomials.
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