x+6=1/6x

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

x+6=1/6x

We move all terms to the left:

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


Domain of the equation: 6x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

x-1/6x+6=0

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2+36x-1=0

a = 6; b = 36; c = -1;
Δ = b2-4ac
Δ = 362-4·6·(-1)
Δ = 1320
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:

$\sqrt{\Delta}=\sqrt{1320}=\sqrt{4*330}=\sqrt{4}*\sqrt{330}=2\sqrt{330}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-2\sqrt{330}}{2*6}=\frac{-36-2\sqrt{330}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+2\sqrt{330}}{2*6}=\frac{-36+2\sqrt{330}}{12}
$