x(x+1)/2=100

Solution for x(x+1)/2=100 equation:

x(x+1)/2=100

We move all terms to the left:

x(x+1)/2-(100)=0

We multiply all the terms by the denominator


x(x+1)-100*2=0

We add all the numbers together, and all the variables

x(x+1)-200=0

We multiply parentheses

x^2+x-200=0

a = 1; b = 1; c = -200;
Δ = b2-4ac
Δ = 12-4·1·(-200)
Δ = 801
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{801}=\sqrt{9*89}=\sqrt{9}*\sqrt{89}=3\sqrt{89}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-3\sqrt{89}}{2*1}=\frac{-1-3\sqrt{89}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+3\sqrt{89}}{2*1}=\frac{-1+3\sqrt{89}}{2}
$