n^2-2n+3=5000

Solution for n^2-2n+3=5000 equation:

n^2-2n+3=5000

We move all terms to the left:

n^2-2n+3-(5000)=0

We add all the numbers together, and all the variables

n^2-2n-4997=0

a = 1; b = -2; c = -4997;
Δ = b2-4ac
Δ = -22-4·1·(-4997)
Δ = 19992
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:

$\sqrt{\Delta}=\sqrt{19992}=\sqrt{196*102}=\sqrt{196}*\sqrt{102}=14\sqrt{102}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-14\sqrt{102}}{2*1}=\frac{2-14\sqrt{102}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+14\sqrt{102}}{2*1}=\frac{2+14\sqrt{102}}{2}
$