5n-10=75/n

Solution for 5n-10=75/n equation:

5n-10=75/n

We move all terms to the left:

5n-10-(75/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

5n-(+75/n)-10=0

We get rid of parentheses

5n-75/n-10=0

We multiply all the terms by the denominator


5n*n-10*n-75=0

We add all the numbers together, and all the variables

-10n+5n*n-75=0

Wy multiply elements

5n^2-10n-75=0

a = 5; b = -10; c = -75;
Δ = b2-4ac
Δ = -102-4·5·(-75)
Δ = 1600
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1600}=40$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-40}{2*5}=\frac{-30}{10}
=-3
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+40}{2*5}=\frac{50}{10}
=5
$