1/5n=2n-18

Solution for 1/5n=2n-18 equation:

1/5n=2n-18

We move all terms to the left:

1/5n-(2n-18)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R


We get rid of parentheses

1/5n-2n+18=0

We multiply all the terms by the denominator


-2n*5n+18*5n+1=0

Wy multiply elements

-10n^2+90n+1=0

a = -10; b = 90; c = +1;
Δ = b2-4ac
Δ = 902-4·(-10)·1
Δ = 8140
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:

$\sqrt{\Delta}=\sqrt{8140}=\sqrt{4*2035}=\sqrt{4}*\sqrt{2035}=2\sqrt{2035}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(90)-2\sqrt{2035}}{2*-10}=\frac{-90-2\sqrt{2035}}{-20}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(90)+2\sqrt{2035}}{2*-10}=\frac{-90+2\sqrt{2035}}{-20}
$