x+1/x-1=6

Solution for x+1/x-1=6 equation:

x+1/x-1=6

We move all terms to the left:

x+1/x-1-(6)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x+1/x-7=0

We multiply all the terms by the denominator


x*x-7*x+1=0

We add all the numbers together, and all the variables

-7x+x*x+1=0

Wy multiply elements

x^2-7x+1=0

a = 1; b = -7; c = +1;
Δ = b2-4ac
Δ = -72-4·1·1
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-3\sqrt{5}}{2*1}=\frac{7-3\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+3\sqrt{5}}{2*1}=\frac{7+3\sqrt{5}}{2}
$