3^x-5=1/9^x

Solution for 3^x-5=1/9^x equation:

3^x-5=1/9^x

We move all terms to the left:

3^x-5-(1/9^x)=0


Domain of the equation: 9^x)!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

3^x-1/9^x-5=0

We multiply all the terms by the denominator


3^x*9^x-5*9^x-1=0

Wy multiply elements

27x^2-45x-1=0

a = 27; b = -45; c = -1;
Δ = b2-4ac
Δ = -452-4·27·(-1)
Δ = 2133
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:

$\sqrt{\Delta}=\sqrt{2133}=\sqrt{9*237}=\sqrt{9}*\sqrt{237}=3\sqrt{237}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-45)-3\sqrt{237}}{2*27}=\frac{45-3\sqrt{237}}{54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-45)+3\sqrt{237}}{2*27}=\frac{45+3\sqrt{237}}{54}
$