8n(n+1)=1

Solution for 8n(n+1)=1 equation:

8n(n+1)=1

We move all terms to the left:

8n(n+1)-(1)=0

We multiply parentheses

8n^2+8n-1=0

a = 8; b = 8; c = -1;
Δ = b2-4ac
Δ = 82-4·8·(-1)
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-4\sqrt{6}}{2*8}=\frac{-8-4\sqrt{6}}{16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+4\sqrt{6}}{2*8}=\frac{-8+4\sqrt{6}}{16}
$