3^n-1/9^1-n=1

Solution for 3^n-1/9^1-n=1 equation:

3^n-1/9^1-n=1

We move all terms to the left:

3^n-1/9^1-n-(1)=0

determiningTheFunctionDomain
3^n-n-1-1/9^1=0

We add all the numbers together, and all the variables

-1n+3^n-1-1/9^1=0

We multiply all the terms by the denominator


-1n*9^1+3^n*9^1-1-1*9^1=0

We add all the numbers together, and all the variables

-1n*9^1+3^n*9^1-10=0

Wy multiply elements

-9n^2+27n^2-10=0

We add all the numbers together, and all the variables

18n^2-10=0

a = 18; b = 0; c = -10;
Δ = b2-4ac
Δ = 02-4·18·(-10)
Δ = 720
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{720}=\sqrt{144*5}=\sqrt{144}*\sqrt{5}=12\sqrt{5}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{5}}{2*18}=\frac{0-12\sqrt{5}}{36}
=-\frac{12\sqrt{5}}{36}
=-\frac{\sqrt{5}}{3}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{5}}{2*18}=\frac{0+12\sqrt{5}}{36}
=\frac{12\sqrt{5}}{36}
=\frac{\sqrt{5}}{3}
$