1/6n-3n-12=2n+13

Solution for 1/6n-3n-12=2n+13 equation:

1/6n-3n-12=2n+13

We move all terms to the left:

1/6n-3n-12-(2n+13)=0


Domain of the equation: 6n!=0

n!=0/6

n!=0

n∈R


We add all the numbers together, and all the variables

-3n+1/6n-(2n+13)-12=0

We get rid of parentheses

-3n+1/6n-2n-13-12=0

We multiply all the terms by the denominator


-3n*6n-2n*6n-13*6n-12*6n+1=0

Wy multiply elements

-18n^2-12n^2-78n-72n+1=0

We add all the numbers together, and all the variables

-30n^2-150n+1=0

a = -30; b = -150; c = +1;
Δ = b2-4ac
Δ = -1502-4·(-30)·1
Δ = 22620
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:

$\sqrt{\Delta}=\sqrt{22620}=\sqrt{4*5655}=\sqrt{4}*\sqrt{5655}=2\sqrt{5655}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-150)-2\sqrt{5655}}{2*-30}=\frac{150-2\sqrt{5655}}{-60}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-150)+2\sqrt{5655}}{2*-30}=\frac{150+2\sqrt{5655}}{-60}
$