1/3x=120-x

Solution for 1/3x=120-x equation:

1/3x=120-x

We move all terms to the left:

1/3x-(120-x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

1/3x-(-1x+120)=0

We get rid of parentheses

1/3x+1x-120=0

We multiply all the terms by the denominator


1x*3x-120*3x+1=0

Wy multiply elements

3x^2-360x+1=0

a = 3; b = -360; c = +1;
Δ = b2-4ac
Δ = -3602-4·3·1
Δ = 129588
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:

$\sqrt{\Delta}=\sqrt{129588}=\sqrt{4*32397}=\sqrt{4}*\sqrt{32397}=2\sqrt{32397}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-360)-2\sqrt{32397}}{2*3}=\frac{360-2\sqrt{32397}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-360)+2\sqrt{32397}}{2*3}=\frac{360+2\sqrt{32397}}{6}
$