2-1/5n=3n+16

Solution for 2-1/5n=3n+16 equation:

2-1/5n=3n+16

We move all terms to the left:

2-1/5n-(3n+16)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R


We get rid of parentheses

-1/5n-3n-16+2=0

We multiply all the terms by the denominator


-3n*5n-16*5n+2*5n-1=0

Wy multiply elements

-15n^2-80n+10n-1=0

We add all the numbers together, and all the variables

-15n^2-70n-1=0

a = -15; b = -70; c = -1;
Δ = b2-4ac
Δ = -702-4·(-15)·(-1)
Δ = 4840
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:

$\sqrt{\Delta}=\sqrt{4840}=\sqrt{484*10}=\sqrt{484}*\sqrt{10}=22\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-70)-22\sqrt{10}}{2*-15}=\frac{70-22\sqrt{10}}{-30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-70)+22\sqrt{10}}{2*-15}=\frac{70+22\sqrt{10}}{-30}
$