125^x=1/25^x-3

Solution for 125^x=1/25^x-3 equation:

125^x=1/25^x-3

We move all terms to the left:

125^x-(1/25^x-3)=0


Domain of the equation: 25^x-3)!=0

x∈R


We get rid of parentheses

125^x-1/25^x+3=0

We multiply all the terms by the denominator


125^x*25^x+3*25^x-1=0

Wy multiply elements

3125x^2+75x-1=0

a = 3125; b = 75; c = -1;
Δ = b2-4ac
Δ = 752-4·3125·(-1)
Δ = 18125
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:

$\sqrt{\Delta}=\sqrt{18125}=\sqrt{625*29}=\sqrt{625}*\sqrt{29}=25\sqrt{29}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(75)-25\sqrt{29}}{2*3125}=\frac{-75-25\sqrt{29}}{6250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(75)+25\sqrt{29}}{2*3125}=\frac{-75+25\sqrt{29}}{6250}
$