1/3x-5+23=x

Solution for 1/3x-5+23=x equation:

1/3x-5+23=x

We move all terms to the left:

1/3x-5+23-(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

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

We multiply all the terms by the denominator


-1x*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 = 2928
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:

$\sqrt{\Delta}=\sqrt{2928}=\sqrt{16*183}=\sqrt{16}*\sqrt{183}=4\sqrt{183}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(54)-4\sqrt{183}}{2*-3}=\frac{-54-4\sqrt{183}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(54)+4\sqrt{183}}{2*-3}=\frac{-54+4\sqrt{183}}{-6}
$