x(x+1)+(x+2)=69

Solution for x(x+1)+(x+2)=69 equation:

x(x+1)+(x+2)=69

We move all terms to the left:

x(x+1)+(x+2)-(69)=0

We multiply parentheses

x^2+x+(x+2)-69=0

We get rid of parentheses

x^2+x+x+2-69=0

We add all the numbers together, and all the variables

x^2+2x-67=0

a = 1; b = 2; c = -67;
Δ = b2-4ac
Δ = 22-4·1·(-67)
Δ = 272
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:

$\sqrt{\Delta}=\sqrt{272}=\sqrt{16*17}=\sqrt{16}*\sqrt{17}=4\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2)-4\sqrt{17}}{2*1}=\frac{-2-4\sqrt{17}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2)+4\sqrt{17}}{2*1}=\frac{-2+4\sqrt{17}}{2}
$