Mathematics - Quadratic Equation Question with Solution | TestHub

The polynomial $x^3-3x^2-9x+c$ can be written in the form of $(x-\alpha {)}^2(x-\beta )$ if the equation $x^3-3x^2-9x+c=0$ has two equal roots. Let these be $\alpha ,\alpha ,\beta$. We have $\alpha +\alpha +\beta =3$ or $2\alpha +\beta =3$ ..........(1)$\alpha \alpha +\alpha \beta +\alpha \beta =-9\text{ or }2\alpha \beta +{\alpha }^2=-\mathrm{9..}\dots \dots \dots \cdot \left(2\right)$ Putting value of $\beta$ in (2) we get $2\alpha (3-2\alpha )+{\alpha }^2=-9$ or $6\alpha -3{\alpha }^2=-9$ $\Rightarrow {\alpha }^2-2\alpha -3=0$ $\Rightarrow (\alpha -3)(\alpha +1)=0\Rightarrow \alpha =-1,3$ When $\alpha =-1,\beta =5$ and when $\alpha =3,\beta =-3$. We also have ${\alpha }^2\beta =-\mathrm{c}$ When $\alpha =-1,\beta =5,c=-5$ when $\alpha =3,\beta =-3,c=27$