1/2n-2/3=1/5n-2

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

1/2n-2/3=1/5n-2

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R



Domain of the equation: 5n-2)!=0

n∈R


We get rid of parentheses

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

We calculate fractions


(-100n^2)/90n^2+45n/90n^2+(-18n)/90n^2+2=0

We multiply all the terms by the denominator


(-100n^2)+45n+(-18n)+2*90n^2=0

Wy multiply elements

(-100n^2)+180n^2+45n+(-18n)=0

We get rid of parentheses

-100n^2+180n^2+45n-18n=0

We add all the numbers together, and all the variables

80n^2+27n=0

a = 80; b = 27; c = 0;
Δ = b2-4ac
Δ = 272-4·80·0
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(27)-27}{2*80}=\frac{-54}{160}
=-27/80
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(27)+27}{2*80}=\frac{0}{160}
=0
$