x+1/x=8

Solution for x+1/x=8 equation:

x+1/x=8

We move all terms to the left:

x+1/x-(8)=0


Domain of the equation: x!=0

x∈R


We multiply all the terms by the denominator


x*x-8*x+1=0

We add all the numbers together, and all the variables

-8x+x*x+1=0

Wy multiply elements

x^2-8x+1=0

a = 1; b = -8; c = +1;
Δ = b2-4ac
Δ = -82-4·1·1
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{15}}{2*1}=\frac{8-2\sqrt{15}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{15}}{2*1}=\frac{8+2\sqrt{15}}{2}
$