6n(31n-18)=12(n-1)+3n

Solution for 6n(31n-18)=12(n-1)+3n equation:

6n(31n-18)=12(n-1)+3n

We move all terms to the left:

6n(31n-18)-(12(n-1)+3n)=0

We multiply parentheses

186n^2-108n-(12(n-1)+3n)=0


We calculate terms in parentheses: -(12(n-1)+3n), so:

12(n-1)+3n

We add all the numbers together, and all the variables

3n+12(n-1)

We multiply parentheses

3n+12n-12

We add all the numbers together, and all the variables

15n-12

Back to the equation:

-(15n-12)


We get rid of parentheses

186n^2-108n-15n+12=0

We add all the numbers together, and all the variables

186n^2-123n+12=0

a = 186; b = -123; c = +12;
Δ = b2-4ac
Δ = -1232-4·186·12
Δ = 6201
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:

$\sqrt{\Delta}=\sqrt{6201}=\sqrt{9*689}=\sqrt{9}*\sqrt{689}=3\sqrt{689}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-123)-3\sqrt{689}}{2*186}=\frac{123-3\sqrt{689}}{372}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-123)+3\sqrt{689}}{2*186}=\frac{123+3\sqrt{689}}{372}
$