Question

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

Answer 1

$ x^4-4 x^3+6 x^2-4 x+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 ^2-2 x +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 $
$=45.95 \Rightarrow [ p ]=45$