15n-1/6n=1/6n-1

Solution for 15n-1/6n=1/6n-1 equation:

15n-1/6n=1/6n-1

We move all terms to the left:

15n-1/6n-(1/6n-1)=0


Domain of the equation: 6n!=0

n!=0/6

n!=0

n∈R



Domain of the equation: 6n-1)!=0

n∈R


We get rid of parentheses

15n-1/6n-1/6n+1=0

We multiply all the terms by the denominator


15n*6n+1*6n-1-1=0

We add all the numbers together, and all the variables

15n*6n+1*6n-2=0

Wy multiply elements

90n^2+6n-2=0

a = 90; b = 6; c = -2;
Δ = b2-4ac
Δ = 62-4·90·(-2)
Δ = 756
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:

$\sqrt{\Delta}=\sqrt{756}=\sqrt{36*21}=\sqrt{36}*\sqrt{21}=6\sqrt{21}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-6\sqrt{21}}{2*90}=\frac{-6-6\sqrt{21}}{180}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+6\sqrt{21}}{2*90}=\frac{-6+6\sqrt{21}}{180}
$