125^x-1=1/25^x

Solution for 125^x-1=1/25^x equation:

125^x-1=1/25^x

We move all terms to the left:

125^x-1-(1/25^x)=0


Domain of the equation: 25^x)!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

125^x-1/25^x-1=0

We multiply all the terms by the denominator


125^x*25^x-1*25^x-1=0

Wy multiply elements

3125x^2-25x-1=0

a = 3125; b = -25; c = -1;
Δ = b2-4ac
Δ = -252-4·3125·(-1)
Δ = 13125
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{13125}=\sqrt{625*21}=\sqrt{625}*\sqrt{21}=25\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-25)-25\sqrt{21}}{2*3125}=\frac{25-25\sqrt{21}}{6250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-25)+25\sqrt{21}}{2*3125}=\frac{25+25\sqrt{21}}{6250}
$