(10/n)-6=n

Solution for (10/n)-6=n equation:

(10/n)-6=n

We move all terms to the left:

(10/n)-6-(n)=0


Domain of the equation: n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

(+10/n)-n-6=0

We add all the numbers together, and all the variables

-1n+(+10/n)-6=0

We get rid of parentheses

-1n+10/n-6=0

We multiply all the terms by the denominator


-1n*n-6*n+10=0

We add all the numbers together, and all the variables

-6n-1n*n+10=0

Wy multiply elements

-1n^2-6n+10=0

a = -1; b = -6; c = +10;
Δ = b2-4ac
Δ = -62-4·(-1)·10
Δ = 76
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{19}}{2*-1}=\frac{6-2\sqrt{19}}{-2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{19}}{2*-1}=\frac{6+2\sqrt{19}}{-2}
$