3^x+1+18/3^x=29

Solution for 3^x+1+18/3^x=29 equation:

3^x+1+18/3^x=29

We move all terms to the left:

3^x+1+18/3^x-(29)=0


Domain of the equation: 3^x!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

3^x+18/3^x-28=0

We multiply all the terms by the denominator


3^x*3^x-28*3^x+18=0

Wy multiply elements

9x^2-84x+18=0

a = 9; b = -84; c = +18;
Δ = b2-4ac
Δ = -842-4·9·18
Δ = 6408
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:

$\sqrt{\Delta}=\sqrt{6408}=\sqrt{36*178}=\sqrt{36}*\sqrt{178}=6\sqrt{178}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-6\sqrt{178}}{2*9}=\frac{84-6\sqrt{178}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+6\sqrt{178}}{2*9}=\frac{84+6\sqrt{178}}{18}
$