x-13=1/3x+5

Solution for x-13=1/3x+5 equation:

x-13=1/3x+5

We move all terms to the left:

x-13-(1/3x+5)=0


Domain of the equation: 3x+5)!=0

x∈R


We get rid of parentheses

x-1/3x-5-13=0

We multiply all the terms by the denominator


x*3x-5*3x-13*3x-1=0

Wy multiply elements

3x^2-15x-39x-1=0

We add all the numbers together, and all the variables

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}
$