Question

If $1,\alpha ,\alpha$ are the roots of the equation $x^3-6x^2+9x-4=0$, then find the value of $\alpha$.

Answer 1

Fact: If $\alpha ,\beta ,\gamma$ are the roots of cubic equation $a{x}^3+b{x}^2+cx+d=0$, then

$\alpha +\beta +\gamma =-\frac{b}{a}$

Thus if $1,\alpha ,\alpha$ are the roots of $x^3-6x^2+9x-4=0$, then

$1+\alpha +\alpha =-\frac{-6}{1}=6$

$\Rightarrow 2\alpha =5$

$\therefore \alpha =\frac{5}{2}$