n*2^n=10000

Solution for n*2^n=10000 equation:

n*2^n=10000

We move all terms to the left:

n*2^n-(10000)=0

Wy multiply elements

2n^2-10000=0

a = 2; b = 0; c = -10000;
Δ = b2-4ac
Δ = 02-4·2·(-10000)
Δ = 80000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{80000}=\sqrt{40000*2}=\sqrt{40000}*\sqrt{2}=200\sqrt{2}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-200\sqrt{2}}{2*2}=\frac{0-200\sqrt{2}}{4}
=-\frac{200\sqrt{2}}{4}
=-50\sqrt{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+200\sqrt{2}}{2*2}=\frac{0+200\sqrt{2}}{4}
=\frac{200\sqrt{2}}{4}
=50\sqrt{2}
$