If $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are zeroes of the polynomial $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-5{\mathrm{x}}^2-2\mathrm{x}+24$, and $\mathrm{\alpha \beta }=12$, then find  $\mathrm{\alpha },\mathrm{\beta }$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-5{\mathrm{x}}^2-2\mathrm{x}+24\to \mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma } \\ \mathrm{Given}, \mathrm{\alpha \beta }=12 \\ \mathrm{products} \mathrm{of} \mathrm{zeroes} \mathrm{\alpha }\mathrm{\beta }\mathrm{\gamma }=-24 \\ 12\mathrm{\gamma }=-24 \\ \mathrm{\gamma }=-2 \\ \mathrm{sum} \mathrm{of} \mathrm{zeroes} \mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=5 \\ \mathrm{\alpha }+\mathrm{\beta }-2=5 \\ \mathrm{\alpha }+\mathrm{\beta }=7 \\ {\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2={\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-4\mathrm{\alpha \beta } \\ \therefore {\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2=49-48 \\ (\mathrm{\alpha }-\mathrm{\beta }{)}^2=1 \\ \begin{array}{c}\mathrm{\alpha }-\mathrm{\beta }=1\\ \mathrm{\alpha }+\mathrm{\beta }=7\end{array} \\ \mathrm{\_\_\_\_\_\_\_\_\_} \\ 2\mathrm{\alpha }=8 \\ \begin{array}{r}\mathrm{\alpha }=4\\ \mathrm{\beta }=3\end{array} \end{gathered}$$