5^n*5^n+4=25

Solution for 5^n*5^n+4=25 equation:

5^n*5^n+4=25

We move all terms to the left:

5^n*5^n+4-(25)=0

We add all the numbers together, and all the variables

5^n*5^n-21=0

Wy multiply elements

25n^2-21=0

a = 25; b = 0; c = -21;
Δ = b2-4ac
Δ = 02-4·25·(-21)
Δ = 2100
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:

$\sqrt{\Delta}=\sqrt{2100}=\sqrt{100*21}=\sqrt{100}*\sqrt{21}=10\sqrt{21}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-10\sqrt{21}}{2*25}=\frac{0-10\sqrt{21}}{50}
=-\frac{10\sqrt{21}}{50}
=-\frac{\sqrt{21}}{5}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+10\sqrt{21}}{2*25}=\frac{0+10\sqrt{21}}{50}
=\frac{10\sqrt{21}}{50}
=\frac{\sqrt{21}}{5}
$