(n(n-1))/2=45

Solution for (n(n-1))/2=45 equation:

(n(n-1))/2=45

We move all terms to the left:

(n(n-1))/2-(45)=0

We multiply all the terms by the denominator


(n(n-1))-45*2=0


We calculate terms in parentheses: +(n(n-1)), so:

n(n-1)

We multiply parentheses

n^2-1n

Back to the equation:

+(n^2-1n)


We add all the numbers together, and all the variables

(n^2-1n)-90=0

We get rid of parentheses

n^2-1n-90=0

a = 1; b = -1; c = -90;
Δ = b2-4ac
Δ = -12-4·1·(-90)
Δ = 361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{361}=19$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-19}{2*1}=\frac{-18}{2}
=-9
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+19}{2*1}=\frac{20}{2}
=10
$