3^x+3+1/3^x-28=0

Solution for 3^x+3+1/3^x-28=0 equation:

3^x+3+1/3^x-28=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+1/3^x-25=0

We multiply all the terms by the denominator


3^x*3^x-25*3^x+1=0

Wy multiply elements

9x^2-75x+1=0

a = 9; b = -75; c = +1;
Δ = b2-4ac
Δ = -752-4·9·1
Δ = 5589
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:

$\sqrt{\Delta}=\sqrt{5589}=\sqrt{81*69}=\sqrt{81}*\sqrt{69}=9\sqrt{69}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-75)-9\sqrt{69}}{2*9}=\frac{75-9\sqrt{69}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-75)+9\sqrt{69}}{2*9}=\frac{75+9\sqrt{69}}{18}
$