recall the quadratic formula is
[tex]\bf ~~~~~~~~~~~~\textit{quadratic formula}
\\\\
y=\stackrel{\stackrel{a}{\downarrow }}{a}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c}
\qquad \qquad
x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}[/tex]
and the tell-tale part is the discriminant, namely the radicand above,
[tex]\bf \qquad \qquad \qquad \textit{discriminant of a quadratic}
\\\\\\
y=\stackrel{\stackrel{a}{\downarrow }}{a}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c}
~~~~~~~~
\stackrel{discriminant}{b^2-4ac}=
\begin{cases}
0&\textit{one solution}\\
positive&\textit{two solutions}\\
negative&\textit{no solution}
\end{cases}[/tex]
so, we can check its discriminant, and if it's negative, then we know it has no solutions, if is positive, then we know we can factor it.
lets check it anyway.
[tex]\bf \qquad \qquad \qquad \textit{discriminant of a quadratic}
\\\\\\
\stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{+7}x\stackrel{\stackrel{c}{\downarrow }}{+3}=0 ~~~~~~~~
\stackrel{discriminant}{b^2-4ac}=
\begin{cases}
0&\textit{one solution}\\
positive&\textit{two solutions}\\
negative&\textit{no solution}
\end{cases}
\\\\\\
(7)^2-4(2)(3)\implies 49-24\implies 25\impliedby \textit{is positive}[/tex]
[tex]\bf ~~~~~~~~~~~~\textit{quadratic formula}
\\\\
\stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{+7}x\stackrel{\stackrel{c}{\downarrow }}{+3}=0
\qquad \qquad
x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}
\\\\\\
x=\cfrac{-7\pm\sqrt{25}}{2(2)}\implies x=\cfrac{-7\pm 5}{4}\implies x=
\begin{cases}
-\cfrac{2}{4}\implies &-\cfrac{1}{2}\\\\
\cfrac{-12}{4}\implies &-3
\end{cases}[/tex]
