Question

ifα,β are two roots of x^2+x+1=0 then find α/β+β/α​

Answer 1

$\bold\red{\underline{\underline{Answer:}}}$

$\bold{\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=-1}$

$\bold\green{\underline{\underline{Solution}}}$

The given quadratic equation is

$\bold{x^{2}+x+1=0}$

Here,

a=1,b=1 and c=1

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

$\bold{\alpha+\beta=-1}$

On squaring both the sides

$\bold{(\alpha+\beta)^{2}=1...(1)}$

_____________________________________

$\bold{\alpha×\beta=\frac{c}{a}}$

$\bold{\alpha×\beta=1...(2)}$

_____________________________________

$\bold{\frac{\alpha}{\beta}+\frac{\beta}{\alpha}}$

$\bold{=\frac{\alpha^{2}+\beta^{2}}{\alpha×\beta}}$

$\bold{=\frac{(\alpha+\beta)^{2}-2(\alpha×\beta)}{(\alpha×\beta)}}$

$\bold{=\frac{1-2}{1}}$

--------From (1) and (2)

=-1

Therefore,

$\bold\purple{\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=-1}$