## Intermediate Algebra: Connecting Concepts through Application

$\left( 3m^{\frac{1}{2}}-2 \right)\left(m^{\frac{1}{2}} - 7 \right)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $3m-23m^{\frac{1}{2}}+14 ,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $3(14)=42$ and the value of $b$ is $-23 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -2,-21 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 3m-2m^{\frac{1}{2}}-21m^{\frac{1}{2}}+14 .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} \left( 3m-2m^{\frac{1}{2}} \right) - \left( 21m^{\frac{1}{2}}-14 \right) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} m^{\frac{1}{2}}\left( 3m^{\frac{1}{2}}-2 \right) - 7\left( 3m^{\frac{1}{2}}-2 \right) .\end{array} Factoring the $GCF= \left( 3m^{\frac{1}{2}}-2 \right)$ of the entire expression above results to \begin{array}{l}\require{cancel} \left( 3m^{\frac{1}{2}}-2 \right)\left(m^{\frac{1}{2}} - 7 \right) .\end{array}