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

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

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

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

2/3x-(-1x+8)-6=0

We get rid of parentheses

2/3x+1x-8-6=0

We multiply all the terms by the denominator


1x*3x-8*3x-6*3x+2=0

Wy multiply elements

3x^2-24x-18x+2=0

We add all the numbers together, and all the variables

3x^2-42x+2=0

a = 3; b = -42; c = +2;
Δ = b2-4ac
Δ = -422-4·3·2
Δ = 1740
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:

$\sqrt{\Delta}=\sqrt{1740}=\sqrt{4*435}=\sqrt{4}*\sqrt{435}=2\sqrt{435}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{435}}{2*3}=\frac{42-2\sqrt{435}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{435}}{2*3}=\frac{42+2\sqrt{435}}{6}
$