1/3x+4+12x-5=22/3x+1

Solution for 1/3x+4+12x-5=22/3x+1 equation:

1/3x+4+12x-5=22/3x+1

We move all terms to the left:

1/3x+4+12x-5-(22/3x+1)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x+1)!=0

x∈R


We add all the numbers together, and all the variables

12x+1/3x-(22/3x+1)-1=0

We get rid of parentheses

12x+1/3x-22/3x-1-1=0

We multiply all the terms by the denominator


12x*3x-1*3x-1*3x+1-22=0

We add all the numbers together, and all the variables

12x*3x-1*3x-1*3x-21=0

Wy multiply elements

36x^2-3x-3x-21=0

We add all the numbers together, and all the variables

36x^2-6x-21=0

a = 36; b = -6; c = -21;
Δ = b2-4ac
Δ = -62-4·36·(-21)
Δ = 3060
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:

$\sqrt{\Delta}=\sqrt{3060}=\sqrt{36*85}=\sqrt{36}*\sqrt{85}=6\sqrt{85}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6\sqrt{85}}{2*36}=\frac{6-6\sqrt{85}}{72}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6\sqrt{85}}{2*36}=\frac{6+6\sqrt{85}}{72}
$