x(6+x)=84

Solution for x(6+x)=84 equation:

x(6+x)=84

We move all terms to the left:

x(6+x)-(84)=0

We add all the numbers together, and all the variables

x(x+6)-84=0

We multiply parentheses

x^2+6x-84=0

a = 1; b = 6; c = -84;
Δ = b2-4ac
Δ = 62-4·1·(-84)
Δ = 372
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:

$\sqrt{\Delta}=\sqrt{372}=\sqrt{4*93}=\sqrt{4}*\sqrt{93}=2\sqrt{93}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{93}}{2*1}=\frac{-6-2\sqrt{93}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{93}}{2*1}=\frac{-6+2\sqrt{93}}{2}
$