125^x=1/25^x

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

125^x=1/25^x

We move all terms to the left:

125^x-(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=0

We multiply all the terms by the denominator


125^x*25^x-1=0

Wy multiply elements

3125x^2-1=0

a = 3125; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·3125·(-1)
Δ = 12500
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:

$\sqrt{\Delta}=\sqrt{12500}=\sqrt{2500*5}=\sqrt{2500}*\sqrt{5}=50\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-50\sqrt{5}}{2*3125}=\frac{0-50\sqrt{5}}{6250}
=-\frac{50\sqrt{5}}{6250}
=-\frac{\sqrt{5}}{125}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+50\sqrt{5}}{2*3125}=\frac{0+50\sqrt{5}}{6250}
=\frac{50\sqrt{5}}{6250}
=\frac{\sqrt{5}}{125}
$