125^x+5=1/25^x

Solution for 125^x+5=1/25^x equation:

125^x+5=1/25^x

We move all terms to the left:

125^x+5-(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+5=0

We multiply all the terms by the denominator


125^x*25^x+5*25^x-1=0

Wy multiply elements

3125x^2+125x-1=0

a = 3125; b = 125; c = -1;
Δ = b2-4ac
Δ = 1252-4·3125·(-1)
Δ = 28125
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:

$\sqrt{\Delta}=\sqrt{28125}=\sqrt{5625*5}=\sqrt{5625}*\sqrt{5}=75\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(125)-75\sqrt{5}}{2*3125}=\frac{-125-75\sqrt{5}}{6250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(125)+75\sqrt{5}}{2*3125}=\frac{-125+75\sqrt{5}}{6250}
$