x+1/3x=18

Solution for x+1/3x=18 equation:

x+1/3x=18

We move all terms to the left:

x+1/3x-(18)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We multiply all the terms by the denominator


x*3x-18*3x+1=0

Wy multiply elements

3x^2-54x+1=0

a = 3; b = -54; c = +1;
Δ = b2-4ac
Δ = -542-4·3·1
Δ = 2904
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:

$\sqrt{\Delta}=\sqrt{2904}=\sqrt{484*6}=\sqrt{484}*\sqrt{6}=22\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-22\sqrt{6}}{2*3}=\frac{54-22\sqrt{6}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+22\sqrt{6}}{2*3}=\frac{54+22\sqrt{6}}{6}
$