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Factoring Quadratics

Factoring Quadratics 1.23 - Solution

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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. 6x221x\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 6{\color{#0000FF}{6}} and 21.{\color{#0000FF}{21}}. Factors of 6: 1,2,3, and 6Factors of 21: 1,3,7, and 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 3.{\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. Factors of 1st variable: x,x2Factors of 2nd variable: x\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, x.{\color{#FF0000}{x}}. Therefore, the GCF of the expression is 3x=3x.{\color{#FF0000}{3}}\cdot{\color{#FF0000}{x}}={\color{#FF0000}{3x}}. Now, we can write the given expression in terms of the GCF. 6x221x3x2x3x7\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. 3x2x3x73x(2x7)\begin{gathered} {\color{#FF0000}{3x}}\cdot 2x-{\color{#FF0000}{3x}}\cdot 7\quad\Leftrightarrow\quad{\color{#FF0000}{3x}}(2x-7) \end{gathered}