Question

Let $p$ be the product of the non real roots of the equation $x^4-4x^3+6x^2-4x=2008$.
where $\left[{}^{\ast }\right]$ denotes the greatest integer function, then find $[p]$

Answer 1

Answer: 45

$x^4-4x^3+6x^2-4x+1=2009$
$(x-1{)}^4=2009$
$(x-1{)}^2=\pm \sqrt{2009}$
as we want only the product of non real roots,
hence, $(x-1{)}^2=-\sqrt{2009}=-7\sqrt{41}$
$x^2-2x+1+7\sqrt{41}=0$
$\Rightarrow \text{ product of roots }=1+7\sqrt{41}$
$\therefore p=1+7\sqrt{41}$
$=1+7(6.42)$
$=1+44.94$
$=45.95\Rightarrow [p]=45$