3^x=3/2x+3

Solution for 3^x=3/2x+3 equation:

3^x=3/2x+3

We move all terms to the left:

3^x-(3/2x+3)=0


Domain of the equation: 2x+3)!=0

x∈R


We get rid of parentheses

3^x-3/2x-3=0

We multiply all the terms by the denominator


3^x*2x-3*2x-3=0

Wy multiply elements

6x^2-6x-3=0

a = 6; b = -6; c = -3;
Δ = b2-4ac
Δ = -62-4·6·(-3)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6\sqrt{3}}{2*6}=\frac{6-6\sqrt{3}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6\sqrt{3}}{2*6}=\frac{6+6\sqrt{3}}{12}
$