x*(x+1)/2+x*(x+1)=30

Solution for x*(x+1)/2+x*(x+1)=30 equation:

x(x+1)/2+x(x+1)=30

We move all terms to the left:

x(x+1)/2+x(x+1)-(30)=0

We multiply parentheses

x^2+x(x+1)/2+x-30=0

We multiply all the terms by the denominator


x^2*2+x(x+1)+x*2-30*2=0

We add all the numbers together, and all the variables

x^2*2+x(x+1)+x*2-60=0

We multiply parentheses

x^2*2+x^2+x+x*2-60=0

Wy multiply elements

2x^2+x^2+x+2x-60=0

We add all the numbers together, and all the variables

3x^2+3x-60=0

a = 3; b = 3; c = -60;
Δ = b2-4ac
Δ = 32-4·3·(-60)
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-27}{2*3}=\frac{-30}{6}
=-5
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+27}{2*3}=\frac{24}{6}
=4
$