x+3/x-3+6(x-3/x+3)=5

Solution for x+3/x-3+6(x-3/x+3)=5 equation:

x+3/x-3+6(x-3/x+3)=5

We move all terms to the left:

x+3/x-3+6(x-3/x+3)-(5)=0


Domain of the equation: x!=0

x∈R



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

x∈R


We add all the numbers together, and all the variables

x+3/x+6(x-3/x+3)-8=0

We multiply parentheses

x+3/x+6x-18x+18-8=0

We multiply all the terms by the denominator


x*x+6x*x-18x*x+18*x-8*x+3=0

We add all the numbers together, and all the variables

10x+x*x+6x*x-18x*x+3=0

Wy multiply elements

x^2+6x^2-18x^2+10x+3=0

We add all the numbers together, and all the variables

-11x^2+10x+3=0

a = -11; b = 10; c = +3;
Δ = b2-4ac
Δ = 102-4·(-11)·3
Δ = 232
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:

$\sqrt{\Delta}=\sqrt{232}=\sqrt{4*58}=\sqrt{4}*\sqrt{58}=2\sqrt{58}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{58}}{2*-11}=\frac{-10-2\sqrt{58}}{-22}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{58}}{2*-11}=\frac{-10+2\sqrt{58}}{-22}
$