3^x+1+18/3^x=27

Solution for 3^x+1+18/3^x=27 equation:

3^x+1+18/3^x=27

We move all terms to the left:

3^x+1+18/3^x-(27)=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-26=0

We multiply all the terms by the denominator


3^x*3^x-26*3^x+18=0

Wy multiply elements

9x^2-78x+18=0

a = 9; b = -78; c = +18;
Δ = b2-4ac
Δ = -782-4·9·18
Δ = 5436
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:

$\sqrt{\Delta}=\sqrt{5436}=\sqrt{36*151}=\sqrt{36}*\sqrt{151}=6\sqrt{151}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-78)-6\sqrt{151}}{2*9}=\frac{78-6\sqrt{151}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-78)+6\sqrt{151}}{2*9}=\frac{78+6\sqrt{151}}{18}
$