x-4=5/3x+2

Solution for x-4=5/3x+2 equation:

x-4=5/3x+2

We move all terms to the left:

x-4-(5/3x+2)=0


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

x∈R


We get rid of parentheses

x-5/3x-2-4=0

We multiply all the terms by the denominator


x*3x-2*3x-4*3x-5=0

Wy multiply elements

3x^2-6x-12x-5=0

We add all the numbers together, and all the variables

3x^2-18x-5=0

a = 3; b = -18; c = -5;
Δ = b2-4ac
Δ = -182-4·3·(-5)
Δ = 384
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:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-8\sqrt{6}}{2*3}=\frac{18-8\sqrt{6}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+8\sqrt{6}}{2*3}=\frac{18+8\sqrt{6}}{6}
$