2/3n=5n-37

Solution for 2/3n=5n-37 equation:

2/3n=5n-37

We move all terms to the left:

2/3n-(5n-37)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

2/3n-5n+37=0

We multiply all the terms by the denominator


-5n*3n+37*3n+2=0

Wy multiply elements

-15n^2+111n+2=0

a = -15; b = 111; c = +2;
Δ = b2-4ac
Δ = 1112-4·(-15)·2
Δ = 12441
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(111)-\sqrt{12441}}{2*-15}=\frac{-111-\sqrt{12441}}{-30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(111)+\sqrt{12441}}{2*-15}=\frac{-111+\sqrt{12441}}{-30}
$