Question

If the sum of two roots of the equation $4x^4-8x^3-13x^2+2x+3=0$ is zero, find largest of all of its roots ?

Answer 1

Let $\alpha ,\beta ,\gamma ,\delta$ are the roots and $\alpha +\beta =0$
$\therefore s_1=\alpha +\beta +\gamma +\delta =-(-\frac{8}{4})=2\Rightarrow \gamma +\delta =2$ ...(1)
$s_2=\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta =-\frac{13}{4}$
$\Rightarrow \alpha \beta +\gamma (\alpha +\beta )+\delta (\alpha +\beta )+\gamma \delta =-\frac{13}{4}\Rightarrow \alpha \beta +\gamma \delta =-\frac{13}{4}$ ...(2)
$s_3=\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta =-\frac{1}{2}$
$\Rightarrow (\alpha +\beta )\gamma \delta +\alpha \beta (\gamma +\delta )=-\frac{1}{2}\Rightarrow \alpha \beta (\gamma +\delta )=-\frac{1}{2}$
$s_4=\alpha \beta \gamma \delta =\frac{3}{4}$ ...(4)
Solving (1),(2),(3) and (4) we get
$\alpha =-\frac{1}{2},\beta =\frac{1}{2},\gamma =-1,\delta =3$