If $2 a+3 b+6 c=0$, then atleast one root of the equation $a x^2+b x+c=0$ lies in the interval

Consider the function $f(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x+d$ We have, $\quad f(0)=d$ and $\quad f(1)=\frac{a}{3}+\frac{b}{2}+c+d$ $\begin{aligned} & =\frac{2 a+3 b+6 c}{6}+d \\ & =\frac{0}{6}+d \quad[\because 2 a+3 b+6 c=0]\end{aligned}$ Therefore, 0 and 1 are roots of the polynomial $f(x)$. Hence, according to Rolle's theorem, there exists atleast one root of the polynomial $f^{\prime}(x)=a x^2+b x+c$ lying between 0 and 1.