(x)(x+1)=x+x+1+19

Solution for (x)(x+1)=x+x+1+19 equation:

(x)(x+1)=x+x+1+19

We move all terms to the left:

(x)(x+1)-(x+x+1+19)=0

We add all the numbers together, and all the variables

x(x+1)-(2x+20)=0

We multiply parentheses

x^2+x-(2x+20)=0

We get rid of parentheses

x^2+x-2x-20=0

We add all the numbers together, and all the variables

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