6x=1/3x+6

Solution for 6x=1/3x+6 equation:

6x=1/3x+6

We move all terms to the left:

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


Domain of the equation: 3x+6)!=0

x∈R


We get rid of parentheses

6x-1/3x-6=0

We multiply all the terms by the denominator


6x*3x-6*3x-1=0

Wy multiply elements

18x^2-18x-1=0

a = 18; b = -18; c = -1;
Δ = b2-4ac
Δ = -182-4·18·(-1)
Δ = 396
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:

$\sqrt{\Delta}=\sqrt{396}=\sqrt{36*11}=\sqrt{36}*\sqrt{11}=6\sqrt{11}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-6\sqrt{11}}{2*18}=\frac{18-6\sqrt{11}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+6\sqrt{11}}{2*18}=\frac{18+6\sqrt{11}}{36}
$