[tex]\bf \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{12}t^2\stackrel{\stackrel{b}{\downarrow }}{+4}t\stackrel{\stackrel{c}{\downarrow }}{-9}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\stackrel{rational}{\textit{one solution}}\\ positive&\stackrel{rational}{\textit{\underline{two solutions}}}~~\textit{\Large \checkmark}\\ negative&\textit{no solution} \end{cases} \\\\\\ 4^2-4(12)(-9)\implies 16+432\implies 448[/tex]
