1/2n=3n-15

Solution for 1/2n=3n-15 equation:

1/2n=3n-15

We move all terms to the left:

1/2n-(3n-15)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-3n+15=0

We multiply all the terms by the denominator


-3n*2n+15*2n+1=0

Wy multiply elements

-6n^2+30n+1=0

a = -6; b = 30; c = +1;
Δ = b2-4ac
Δ = 302-4·(-6)·1
Δ = 924
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:

$\sqrt{\Delta}=\sqrt{924}=\sqrt{4*231}=\sqrt{4}*\sqrt{231}=2\sqrt{231}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{231}}{2*-6}=\frac{-30-2\sqrt{231}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{231}}{2*-6}=\frac{-30+2\sqrt{231}}{-12}
$