43-8(n-1)n=1

Solution for 43-8(n-1)n=1 equation:

43-8(n-1)n=1

We move all terms to the left:

43-8(n-1)n-(1)=0

We add all the numbers together, and all the variables

-8(n-1)n+42=0

We multiply parentheses

-8n^2+8n+42=0

a = -8; b = 8; c = +42;
Δ = b2-4ac
Δ = 82-4·(-8)·42
Δ = 1408
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:

$\sqrt{\Delta}=\sqrt{1408}=\sqrt{64*22}=\sqrt{64}*\sqrt{22}=8\sqrt{22}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-8\sqrt{22}}{2*-8}=\frac{-8-8\sqrt{22}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+8\sqrt{22}}{2*-8}=\frac{-8+8\sqrt{22}}{-16}
$