3n*n+1=416

Solution for 3n*n+1=416 equation:

3n*n+1=416

We move all terms to the left:

3n*n+1-(416)=0

We add all the numbers together, and all the variables

3n*n-415=0

Wy multiply elements

3n^2-415=0

a = 3; b = 0; c = -415;
Δ = b2-4ac
Δ = 02-4·3·(-415)
Δ = 4980
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:

$\sqrt{\Delta}=\sqrt{4980}=\sqrt{4*1245}=\sqrt{4}*\sqrt{1245}=2\sqrt{1245}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{1245}}{2*3}=\frac{0-2\sqrt{1245}}{6}
=-\frac{2\sqrt{1245}}{6}
=-\frac{\sqrt{1245}}{3}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{1245}}{2*3}=\frac{0+2\sqrt{1245}}{6}
=\frac{2\sqrt{1245}}{6}
=\frac{\sqrt{1245}}{3}
$