x+(1/3x-6)=90

Solution for x+(1/3x-6)=90 equation:

x+(1/3x-6)=90

We move all terms to the left:

x+(1/3x-6)-(90)=0


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

x∈R


We get rid of parentheses

x+1/3x-6-90=0

We multiply all the terms by the denominator


x*3x-6*3x-90*3x+1=0

Wy multiply elements

3x^2-18x-270x+1=0

We add all the numbers together, and all the variables

3x^2-288x+1=0

a = 3; b = -288; c = +1;
Δ = b2-4ac
Δ = -2882-4·3·1
Δ = 82932
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:

$\sqrt{\Delta}=\sqrt{82932}=\sqrt{4*20733}=\sqrt{4}*\sqrt{20733}=2\sqrt{20733}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-288)-2\sqrt{20733}}{2*3}=\frac{288-2\sqrt{20733}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-288)+2\sqrt{20733}}{2*3}=\frac{288+2\sqrt{20733}}{6}
$