2/3x-1/3x=5/6x+1

Solution for 2/3x-1/3x=5/6x+1 equation:

2/3x-1/3x=5/6x+1

We move all terms to the left:

2/3x-1/3x-(5/6x+1)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 6x+1)!=0

x∈R


We get rid of parentheses

2/3x-1/3x-5/6x-1=0

We calculate fractions


(-6x+2)/18x^2+(-15x)/18x^2-1=0

We multiply all the terms by the denominator


(-6x+2)+(-15x)-1*18x^2=0

Wy multiply elements

-18x^2+(-6x+2)+(-15x)=0

We get rid of parentheses

-18x^2-6x-15x+2=0

We add all the numbers together, and all the variables

-18x^2-21x+2=0

a = -18; b = -21; c = +2;
Δ = b2-4ac
Δ = -212-4·(-18)·2
Δ = 585
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:

$\sqrt{\Delta}=\sqrt{585}=\sqrt{9*65}=\sqrt{9}*\sqrt{65}=3\sqrt{65}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-21)-3\sqrt{65}}{2*-18}=\frac{21-3\sqrt{65}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-21)+3\sqrt{65}}{2*-18}=\frac{21+3\sqrt{65}}{-36}
$