215=8n+(.5n)n

Solution for 215=8n+(.5n)n equation:

215=8n+(.5n)n

We move all terms to the left:

215-(8n+(.5n)n)=0

We add all the numbers together, and all the variables

-(8n+(+.5n)n)+215=0


We calculate terms in parentheses: -(8n+(+.5n)n), so:

8n+(+.5n)n

We multiply parentheses

n^2+8n

Back to the equation:

-(n^2+8n)


We get rid of parentheses

-n^2-8n+215=0

We add all the numbers together, and all the variables

-1n^2-8n+215=0

a = -1; b = -8; c = +215;
Δ = b2-4ac
Δ = -82-4·(-1)·215
Δ = 924
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:

$\sqrt{\Delta}=\sqrt{924}=\sqrt{4*231}=\sqrt{4}*\sqrt{231}=2\sqrt{231}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{231}}{2*-1}=\frac{8-2\sqrt{231}}{-2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{231}}{2*-1}=\frac{8+2\sqrt{231}}{-2}
$