10+3/n=n-6

Solution for 10+3/n=n-6 equation:

10+3/n=n-6

We move all terms to the left:

10+3/n-(n-6)=0


Domain of the equation: n!=0

n∈R


We get rid of parentheses

3/n-n+6+10=0

We multiply all the terms by the denominator


-n*n+6*n+10*n+3=0

We add all the numbers together, and all the variables

16n-n*n+3=0

Wy multiply elements

-1n^2+16n+3=0

a = -1; b = 16; c = +3;
Δ = b2-4ac
Δ = 162-4·(-1)·3
Δ = 268
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:

$\sqrt{\Delta}=\sqrt{268}=\sqrt{4*67}=\sqrt{4}*\sqrt{67}=2\sqrt{67}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-2\sqrt{67}}{2*-1}=\frac{-16-2\sqrt{67}}{-2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+2\sqrt{67}}{2*-1}=\frac{-16+2\sqrt{67}}{-2}
$