Question

If two roots of $x^3+3x^2-9x+c=0$ are equal, then find the value of $c.$

Answer 1

$x^3+3x^2-9x+c=0$

Let three roots $\alpha ,\beta ,\gamma$

$\because 2$ roots are equal $(\alpha =\beta )$

$\Rightarrow roots=\alpha ,\alpha ,\gamma$

$\Rightarrow \alpha +\alpha +\gamma =-3\Rightarrow 2\alpha +\gamma =-3\dots \left(1\right)and{\alpha }^2+\alpha \gamma +\alpha \gamma =-9\Rightarrow {\alpha }^2+2\alpha r=-9and{\alpha }^2\gamma =-c...\left(3\right)$

By solving equation $\left(1\right)$ and $\left(2\right)$

${\alpha }^2+2\alpha -3=0(\alpha -1)(\alpha +3)=0\alpha =\pm 1\Rightarrow \gamma =-5\Rightarrow c=5\alpha =-3\Rightarrow \gamma =3\Rightarrow c=-27$