(n-2)(n-2)=67

Solution for (n-2)(n-2)=67 equation:

(n-2)(n-2)=67

We move all terms to the left:

(n-2)(n-2)-(67)=0

We multiply parentheses ..

(+n^2-2n-2n+4)-67=0

We get rid of parentheses

n^2-2n-2n+4-67=0

We add all the numbers together, and all the variables

n^2-4n-63=0

a = 1; b = -4; c = -63;
Δ = b2-4ac
Δ = -42-4·1·(-63)
Δ = 268
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:

$\sqrt{\Delta}=\sqrt{268}=\sqrt{4*67}=\sqrt{4}*\sqrt{67}=2\sqrt{67}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{67}}{2*1}=\frac{4-2\sqrt{67}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{67}}{2*1}=\frac{4+2\sqrt{67}}{2}
$