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

Solution for (7/x+3)+(5/3x-1)=13/x+3 equation:

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

We move all terms to the left:

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


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

x∈R



Domain of the equation: 3x-1)!=0

x∈R


We get rid of parentheses

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

We calculate fractions


(-39x+7)/3x^2+5x/3x^2+3-1-3=0

We add all the numbers together, and all the variables

(-39x+7)/3x^2+5x/3x^2-1=0

We multiply all the terms by the denominator


(-39x+7)+5x-1*3x^2=0

We add all the numbers together, and all the variables

5x+(-39x+7)-1*3x^2=0

Wy multiply elements

-3x^2+5x+(-39x+7)=0

We get rid of parentheses

-3x^2+5x-39x+7=0

We add all the numbers together, and all the variables

-3x^2-34x+7=0

a = -3; b = -34; c = +7;
Δ = b2-4ac
Δ = -342-4·(-3)·7
Δ = 1240
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:

$\sqrt{\Delta}=\sqrt{1240}=\sqrt{4*310}=\sqrt{4}*\sqrt{310}=2\sqrt{310}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-34)-2\sqrt{310}}{2*-3}=\frac{34-2\sqrt{310}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-34)+2\sqrt{310}}{2*-3}=\frac{34+2\sqrt{310}}{-6}
$