x(x+1)/2=66

Solution for x(x+1)/2=66 equation:

x(x+1)/2=66

We move all terms to the left:

x(x+1)/2-(66)=0

We multiply all the terms by the denominator


x(x+1)-66*2=0

We add all the numbers together, and all the variables

x(x+1)-132=0

We multiply parentheses

x^2+x-132=0

a = 1; b = 1; c = -132;
Δ = b2-4ac
Δ = 12-4·1·(-132)
Δ = 529
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{529}=23$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-23}{2*1}=\frac{-24}{2}
=-12
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+23}{2*1}=\frac{22}{2}
=11
$