1/5^x-7=125^x

Solution for 1/5^x-7=125^x equation:

1/5^x-7=125^x

We move all terms to the left:

1/5^x-7-(125^x)=0


Domain of the equation: 5^x!=0

x!=0/1

x!=0

x∈R


We multiply all the terms by the denominator


-125^x*5^x-7*5^x+1=0

Wy multiply elements

-625x^2-35x+1=0

a = -625; b = -35; c = +1;
Δ = b2-4ac
Δ = -352-4·(-625)·1
Δ = 3725
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:

$\sqrt{\Delta}=\sqrt{3725}=\sqrt{25*149}=\sqrt{25}*\sqrt{149}=5\sqrt{149}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-35)-5\sqrt{149}}{2*-625}=\frac{35-5\sqrt{149}}{-1250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-35)+5\sqrt{149}}{2*-625}=\frac{35+5\sqrt{149}}{-1250}
$