1/3x+13=2x-3

Solution for 1/3x+13=2x-3 equation:

1/3x+13=2x-3

We move all terms to the left:

1/3x+13-(2x-3)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-2x+3+13=0

We multiply all the terms by the denominator


-2x*3x+3*3x+13*3x+1=0

Wy multiply elements

-6x^2+9x+39x+1=0

We add all the numbers together, and all the variables

-6x^2+48x+1=0

a = -6; b = 48; c = +1;
Δ = b2-4ac
Δ = 482-4·(-6)·1
Δ = 2328
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:

$\sqrt{\Delta}=\sqrt{2328}=\sqrt{4*582}=\sqrt{4}*\sqrt{582}=2\sqrt{582}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(48)-2\sqrt{582}}{2*-6}=\frac{-48-2\sqrt{582}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(48)+2\sqrt{582}}{2*-6}=\frac{-48+2\sqrt{582}}{-12}
$