Question

For the equation $3x^2+p+3=0,$ if one of the root is square of the other, then find the value of $p$

Answer 1

Let the roots be $\alpha ,{\alpha }^2$

$\Rightarrow \alpha +{\alpha }^2=\frac{-p}{3}$ ........$\left(1\right)$

${\alpha }^3=1$ ........$\left(2\right)$

From $\left(2\right)\alpha =1\omega ,{\omega }^2$

When $\alpha =1,sum=\alpha +{\alpha }^2=\frac{-p}{3}\Rightarrow 1+1=\frac{-p}{3}$

$\therefore \alpha =\frac{-p}{3}$

When $\alpha ={\omega }^2,{\omega }^2+{\omega }^4=\frac{-p}{3}$

$\Rightarrow p=-6$

$\therefore p=-6$