(x(x+1)/2)=10

Solution for (x(x+1)/2)=10 equation:

(x(x+1)/2)=10

We move all terms to the left:

(x(x+1)/2)-(10)=0

We multiply all the terms by the denominator


(x(x+1)-10*2)=0


We calculate terms in parentheses: +(x(x+1)-10*2), so:

x(x+1)-10*2

We add all the numbers together, and all the variables

x(x+1)-20

We multiply parentheses

x^2+x-20

Back to the equation:

+(x^2+x-20)


We get rid of parentheses

x^2+x-20=0

a = 1; b = 1; c = -20;
Δ = b2-4ac
Δ = 12-4·1·(-20)
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-9}{2*1}=\frac{-10}{2}
=-5
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+9}{2*1}=\frac{8}{2}
=4
$