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*125-78126=0

Wy multiply elements

625n^2-78126=0

a = 625; b = 0; c = -78126;
Δ = b2-4ac
Δ = 02-4·625·(-78126)
Δ = 195315000
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:

$\sqrt{\Delta}=\sqrt{195315000}=\sqrt{2500*78126}=\sqrt{2500}*\sqrt{78126}=50\sqrt{78126}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-50\sqrt{78126}}{2*625}=\frac{0-50\sqrt{78126}}{1250}
=-\frac{50\sqrt{78126}}{1250}
=-\frac{\sqrt{78126}}{25}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+50\sqrt{78126}}{2*625}=\frac{0+50\sqrt{78126}}{1250}
=\frac{50\sqrt{78126}}{1250}
=\frac{\sqrt{78126}}{25}
$