1/3x+4=3x-2=1/2

Solution for 1/3x+4=3x-2=1/2 equation:

1/3x+4=3x-2=1/2

We move all terms to the left:

1/3x+4-(3x-2)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-3x+2+4=0

We multiply all the terms by the denominator


-3x*3x+2*3x+4*3x+1=0

Wy multiply elements

-9x^2+6x+12x+1=0

We add all the numbers together, and all the variables

-9x^2+18x+1=0

a = -9; b = 18; c = +1;
Δ = b2-4ac
Δ = 182-4·(-9)·1
Δ = 360
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:

$\sqrt{\Delta}=\sqrt{360}=\sqrt{36*10}=\sqrt{36}*\sqrt{10}=6\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-6\sqrt{10}}{2*-9}=\frac{-18-6\sqrt{10}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+6\sqrt{10}}{2*-9}=\frac{-18+6\sqrt{10}}{-18}
$