5^x-1/5^x=6

Solution for 5^x-1/5^x=6 equation:

5^x-1/5^x=6

We move all terms to the left:

5^x-1/5^x-(6)=0


Domain of the equation: 5^x!=0

x!=0/1

x!=0

x∈R


We multiply all the terms by the denominator


5^x*5^x-6*5^x-1=0

Wy multiply elements

25x^2-30x-1=0

a = 25; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·25·(-1)
Δ = 1000
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:

$\sqrt{\Delta}=\sqrt{1000}=\sqrt{100*10}=\sqrt{100}*\sqrt{10}=10\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-10\sqrt{10}}{2*25}=\frac{30-10\sqrt{10}}{50}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+10\sqrt{10}}{2*25}=\frac{30+10\sqrt{10}}{50}
$