5^x+1/5^x-1=2

Solution for 5^x+1/5^x-1=2 equation:

5^x+1/5^x-1=2

We move all terms to the left:

5^x+1/5^x-1-(2)=0


Domain of the equation: 5^x!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

5^x+1/5^x-3=0

We multiply all the terms by the denominator


5^x*5^x-3*5^x+1=0

Wy multiply elements

25x^2-15x+1=0

a = 25; b = -15; c = +1;
Δ = b2-4ac
Δ = -152-4·25·1
Δ = 125
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:

$\sqrt{\Delta}=\sqrt{125}=\sqrt{25*5}=\sqrt{25}*\sqrt{5}=5\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-15)-5\sqrt{5}}{2*25}=\frac{15-5\sqrt{5}}{50}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+5\sqrt{5}}{2*25}=\frac{15+5\sqrt{5}}{50}
$