n-5n+12=3n^2

Solution for n-5n+12=3n^2 equation:

n-5n+12=3n^2

We move all terms to the left:

n-5n+12-(3n^2)=0

determiningTheFunctionDomain
-3n^2+n-5n+12=0

We add all the numbers together, and all the variables

-3n^2-4n+12=0

a = -3; b = -4; c = +12;
Δ = b2-4ac
Δ = -42-4·(-3)·12
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-4\sqrt{10}}{2*-3}=\frac{4-4\sqrt{10}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{10}}{2*-3}=\frac{4+4\sqrt{10}}{-6}
$