3/x-4=x-5/x

Solution for 3/x-4=x-5/x equation:

3/x-4=x-5/x

We move all terms to the left:

3/x-4-(x-5/x)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

3/x-(+x-5/x)-4=0

We get rid of parentheses

3/x-x+5/x-4=0

We multiply all the terms by the denominator


-x*x-4*x+3+5=0

We add all the numbers together, and all the variables

-4x-x*x+8=0

Wy multiply elements

-1x^2-4x+8=0

a = -1; b = -4; c = +8;
Δ = b2-4ac
Δ = -42-4·(-1)·8
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{-(-4)-4\sqrt{3}}{2*-1}=\frac{4-4\sqrt{3}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{3}}{2*-1}=\frac{4+4\sqrt{3}}{-2}
$