5^2n+1/125=5^4

Solution for 5^2n+1/125=5^4 equation:

5^2n+1/125=5^4

We move all terms to the left:

5^2n+1/125-(5^4)=0

We add all the numbers together, and all the variables

5^2n-625+1/125=0

We multiply all the terms by the denominator


5^2n*125+1-625*125=0

We add all the numbers together, and all the variables

5^2n*125-78124=0

Wy multiply elements

625n^2-78124=0

a = 625; b = 0; c = -78124;
Δ = b2-4ac
Δ = 02-4·625·(-78124)
Δ = 195310000
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:

$\sqrt{\Delta}=\sqrt{195310000}=\sqrt{10000*19531}=\sqrt{10000}*\sqrt{19531}=100\sqrt{19531}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-100\sqrt{19531}}{2*625}=\frac{0-100\sqrt{19531}}{1250}
=-\frac{100\sqrt{19531}}{1250}
=-\frac{2\sqrt{19531}}{25}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+100\sqrt{19531}}{2*625}=\frac{0+100\sqrt{19531}}{1250}
=\frac{100\sqrt{19531}}{1250}
=\frac{2\sqrt{19531}}{25}
$