3^x=3/2x+2

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

3^x=3/2x+2

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2-4x-3=0

a = 6; b = -4; c = -3;
Δ = b2-4ac
Δ = -42-4·6·(-3)
Δ = 88
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:

$\sqrt{\Delta}=\sqrt{88}=\sqrt{4*22}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{22}}{2*6}=\frac{4-2\sqrt{22}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{22}}{2*6}=\frac{4+2\sqrt{22}}{12}
$