$The condition that sin\theta , cos\theta may be the roots of a{x}^2+bx+c=0 is$

$$\begin{gathered} \sin \theta +\cos \theta =\frac{-b}{a} , \sin \theta \cos \theta =\frac{c}{a} Squaring on both sides \\ {(\sin \theta +\cos \theta )}^2={\left(\frac{-b}{a}\right)}^2 1+2\sin \theta \cos \theta =\frac{b^2}{a^2} \\ 1+2\frac{c}{a} = \frac{b^2}{a^2} \\ \Rightarrow a+2c=\frac{b^2}{a} \\ \Rightarrow a(a+2c)=b^2 \end{gathered}$$