Question

One of the roots of the equation $2000 x^6+100 x^5+10 x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}{r}$, where $m$ is non zero integer and $n$ and $r$ are relatively prime natural numbers. Find the value of $m+n+r$.

Answer 1

$\left(2000 x^6-2\right)+x\left(100 x^4+10 x^2+1\right)=0$
$2\left(\left(10 x^2\right)^3-1\right)+x\left(100 x^4+10 x^2+1\right)=0$
$2\left(10 x^2-1\right)\left(100 x^4+100 x^2+1\right)+x\left(100 x^4+10 x^2+1\right)=0 $
${\left[2\left(10 x^2-1\right)+x\right]\left[100 x^4+10 x^2+1\right]=0} $
$2 x^2+x-2=0 $
$D< 0 \text { no real solution.}$