1/x+5x=10

Solution for 1/x+5x=10 equation:

1/x+5x=10

We move all terms to the left:

1/x+5x-(10)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

5x+1/x-10=0

We multiply all the terms by the denominator


5x*x-10*x+1=0

We add all the numbers together, and all the variables

-10x+5x*x+1=0

Wy multiply elements

5x^2-10x+1=0

a = 5; b = -10; c = +1;
Δ = b2-4ac
Δ = -102-4·5·1
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-4\sqrt{5}}{2*5}=\frac{10-4\sqrt{5}}{10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+4\sqrt{5}}{2*5}=\frac{10+4\sqrt{5}}{10}
$