Question

Given that two roots of $4x^3+20x^2-23x+6=0$ are equal, find all the roots of the given equation.

Answer 1

Let $\alpha ,\beta ,\gamma$ are the roots of

$4x^3+20x^2-23x+6=0$

given two roots are equal, let $\alpha =\beta$

We have $\alpha +\beta +\gamma =\frac{-20}{4}=-5$

$\Rightarrow 2\alpha +\gamma =-5$ .....(1)

$\Rightarrow \alpha \beta +\beta \gamma +\gamma \alpha =\frac{-23}{4}$

$\Rightarrow {\alpha }^2+2\alpha \gamma =\frac{-23}{4}$ .....(2)

$\alpha \beta \gamma =\frac{-6}{4}=\frac{-3}{2}=\Rightarrow {\alpha }^2\gamma =\frac{-3}{2}$

From $\left(1\right)$ and $\left(2\right)$ ${\alpha }^2+2\alpha (-5-2\alpha )$