2x-3=5/6x+3

Solution for 2x-3=5/6x+3 equation:

2x-3=5/6x+3

We move all terms to the left:

2x-3-(5/6x+3)=0


Domain of the equation: 6x+3)!=0

x∈R


We get rid of parentheses

2x-5/6x-3-3=0

We multiply all the terms by the denominator


2x*6x-3*6x-3*6x-5=0

Wy multiply elements

12x^2-18x-18x-5=0

We add all the numbers together, and all the variables

12x^2-36x-5=0

a = 12; b = -36; c = -5;
Δ = b2-4ac
Δ = -362-4·12·(-5)
Δ = 1536
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:

$\sqrt{\Delta}=\sqrt{1536}=\sqrt{256*6}=\sqrt{256}*\sqrt{6}=16\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-16\sqrt{6}}{2*12}=\frac{36-16\sqrt{6}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+16\sqrt{6}}{2*12}=\frac{36+16\sqrt{6}}{24}
$