x(x+4)=109

Solution for x(x+4)=109 equation:

x(x+4)=109

We move all terms to the left:

x(x+4)-(109)=0

We multiply parentheses

x^2+4x-109=0

a = 1; b = 4; c = -109;
Δ = b2-4ac
Δ = 42-4·1·(-109)
Δ = 452
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:

$\sqrt{\Delta}=\sqrt{452}=\sqrt{4*113}=\sqrt{4}*\sqrt{113}=2\sqrt{113}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{113}}{2*1}=\frac{-4-2\sqrt{113}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{113}}{2*1}=\frac{-4+2\sqrt{113}}{2}
$