x(x+1)/2=333

Solution for x(x+1)/2=333 equation:

x(x+1)/2=333

We move all terms to the left:

x(x+1)/2-(333)=0

We multiply all the terms by the denominator


x(x+1)-333*2=0

We add all the numbers together, and all the variables

x(x+1)-666=0

We multiply parentheses

x^2+x-666=0

a = 1; b = 1; c = -666;
Δ = b2-4ac
Δ = 12-4·1·(-666)
Δ = 2665
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{2665}}{2*1}=\frac{-1-\sqrt{2665}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{2665}}{2*1}=\frac{-1+\sqrt{2665}}{2}
$