3x=1/x-6

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

3x=1/x-6

We move all terms to the left:

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


Domain of the equation: x-6)!=0

x∈R


We get rid of parentheses

3x-1/x+6=0

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

6x+3x*x-1=0

Wy multiply elements

3x^2+6x-1=0

a = 3; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·3·(-1)
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-4\sqrt{3}}{2*3}=\frac{-6-4\sqrt{3}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+4\sqrt{3}}{2*3}=\frac{-6+4\sqrt{3}}{6}
$