3x+1=1/2x+3

Solution for 3x+1=1/2x+3 equation:

3x+1=1/2x+3

We move all terms to the left:

3x+1-(1/2x+3)=0


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

x∈R


We get rid of parentheses

3x-1/2x-3+1=0

We multiply all the terms by the denominator


3x*2x-3*2x+1*2x-1=0

Wy multiply elements

6x^2-6x+2x-1=0

We add all the numbers together, and all the variables

6x^2-4x-1=0

a = 6; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·6·(-1)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{10}}{2*6}=\frac{4-2\sqrt{10}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{10}}{2*6}=\frac{4+2\sqrt{10}}{12}
$