(n-3)*n/2=27

Solution for (n-3)*n/2=27 equation:

(n-3)*n/2=27

We move all terms to the left:

(n-3)*n/2-(27)=0

We multiply all the terms by the denominator


(n-3)*n-27*2=0

We add all the numbers together, and all the variables

(n-3)*n-54=0

We multiply parentheses

n^2-3n-54=0

a = 1; b = -3; c = -54;
Δ = b2-4ac
Δ = -32-4·1·(-54)
Δ = 225
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{225}=15$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-15}{2*1}=\frac{-12}{2}
=-6
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+15}{2*1}=\frac{18}{2}
=9
$