125^x=1/5^x-8

Solution for 125^x=1/5^x-8 equation:

125^x=1/5^x-8

We move all terms to the left:

125^x-(1/5^x-8)=0


Domain of the equation: 5^x-8)!=0

x∈R


We get rid of parentheses

125^x-1/5^x+8=0

We multiply all the terms by the denominator


125^x*5^x+8*5^x-1=0

Wy multiply elements

625x^2+40x-1=0

a = 625; b = 40; c = -1;
Δ = b2-4ac
Δ = 402-4·625·(-1)
Δ = 4100
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:

$\sqrt{\Delta}=\sqrt{4100}=\sqrt{100*41}=\sqrt{100}*\sqrt{41}=10\sqrt{41}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-10\sqrt{41}}{2*625}=\frac{-40-10\sqrt{41}}{1250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+10\sqrt{41}}{2*625}=\frac{-40+10\sqrt{41}}{1250}
$