x-1/x-2+x+3/x+4=10

Solution for x-1/x-2+x+3/x+4=10 equation:

x-1/x-2+x+3/x+4=10

We move all terms to the left:

x-1/x-2+x+3/x+4-(10)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

2x-1/x+3/x-8=0

We multiply all the terms by the denominator


2x*x-8*x-1+3=0

We add all the numbers together, and all the variables

-8x+2x*x+2=0

Wy multiply elements

2x^2-8x+2=0

a = 2; b = -8; c = +2;
Δ = b2-4ac
Δ = -82-4·2·2
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{3}}{2*2}=\frac{8-4\sqrt{3}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{3}}{2*2}=\frac{8+4\sqrt{3}}{4}
$