If sin $\alpha$ and cos $\alpha$ are the roots of the equation ax² +bx+c=0, then

. Since, sin $\alpha$ and cos $\alpha$ are the roots of the equation ax²+ bx+ c=0, then,
$\sin \mathrm{\alpha }+\cos \mathrm{\alpha }=-\frac{\mathrm{b}}{\mathrm{a}}\text{ and }\sin \mathrm{\alpha cos}\mathrm{\alpha }=\frac{\mathrm{c}}{\mathrm{a}}$
To eliminate $\alpha$ we have
$1={\sin }^2\mathrm{\alpha }+{\cos }^2\mathrm{\alpha }$
$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1=(\sin \mathrm{\alpha }+\cos \mathrm{\alpha }{)}^2-2\sin \mathrm{\alpha cos}\mathrm{\alpha }\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1=\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}-\frac{2\mathrm{c}}{\mathrm{a}}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}^2-{\mathrm{b}}^2+2\mathrm{ac}=0\end{array}$