Factoring Quadratics

We want to find the greatest common factor — GCF — of the terms in the given expression. To do so, we will consider coefficients and variables separately. $\begin{gathered} {\color{#0000FF}{6}} {\color{#009600}{x^2}}-{\color{#0000FF}{21}} {\color{#009600}{x}} \end{gathered}$ Let's start by finding the GCF of ${\color{#0000FF}{6}}$ and ${\color{#0000FF}{21}}.$ $\begin{aligned} \textbf{Factors of }\mathbf{6}\textbf{:}&\ 1, \, 2, \, {\color{#FF0000}{3}},\text{ and }6\\ \textbf{Factors of }\mathbf{21}\textbf{:}&\ 1, \, {\color{#FF0000}{3}}, \, 7, \text{ and } 21 \end{aligned}$ We found that the GCF of the coefficients is ${\color{#FF0000}{3}}.$ To find the GCF of the variables, we need to identify the variables repeated in both terms, and write them with their minimum exponents. $\begin{aligned} \textbf{Factors of }\mathbf{1^\text{st}}\textbf{ variable:}&\ {\color{#FF0000}{x}}, x^2\\ \textbf{Factors of }\mathbf{2^\text{nd}}\textbf{ variable:}&\ {\color{#FF0000}{x}} \end{aligned}$ We see that there is one repeated variable factor, ${\color{#FF0000}{x}}.$ Therefore, the GCF of the expression is ${\color{#FF0000}{3}}\cdot{\color{#FF0000}{x}}={\color{#FF0000}{3x}}.$ Now, we can write the given expression in terms of the GCF. $\begin{gathered} 6x^2-21x \quad\Leftrightarrow\quad{\color{#FF0000}{3x}}\cdot 2x-{\color{#FF0000}{3x}}\cdot 7 \end{gathered}$ Finally, we will factor out the GCF. $\begin{gathered} {\color{#FF0000}{3x}}\cdot 2x-{\color{#FF0000}{3x}}\cdot 7\quad\Leftrightarrow\quad{\color{#FF0000}{3x}}(2x-7) \end{gathered}$