x=30+1/3x

Solution for x=30+1/3x equation:

x=30+1/3x

We move all terms to the left:

x-(30+1/3x)=0


Domain of the equation: 3x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

x-(1/3x+30)=0

We get rid of parentheses

x-1/3x-30=0

We multiply all the terms by the denominator


x*3x-30*3x-1=0

Wy multiply elements

3x^2-90x-1=0

a = 3; b = -90; c = -1;
Δ = b2-4ac
Δ = -902-4·3·(-1)
Δ = 8112
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:

$\sqrt{\Delta}=\sqrt{8112}=\sqrt{2704*3}=\sqrt{2704}*\sqrt{3}=52\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-90)-52\sqrt{3}}{2*3}=\frac{90-52\sqrt{3}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-90)+52\sqrt{3}}{2*3}=\frac{90+52\sqrt{3}}{6}
$