3x+1=13/x-3

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

3x+1=13/x-3

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

4x+3x*x-13=0

Wy multiply elements

3x^2+4x-13=0

a = 3; b = 4; c = -13;
Δ = b2-4ac
Δ = 42-4·3·(-13)
Δ = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{43}}{2*3}=\frac{-4-2\sqrt{43}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{43}}{2*3}=\frac{-4+2\sqrt{43}}{6}
$