(x+3)2+(x)2=117

Solution for (x+3)2+(x)2=117 equation:

(x+3)2+(x)2=117

We move all terms to the left:

(x+3)2+(x)2-(117)=0

We add all the numbers together, and all the variables

x^2+(x+3)2-117=0

We multiply parentheses

x^2+2x+6-117=0

We add all the numbers together, and all the variables

x^2+2x-111=0

a = 1; b = 2; c = -111;
Δ = b2-4ac
Δ = 22-4·1·(-111)
Δ = 448
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:

$\sqrt{\Delta}=\sqrt{448}=\sqrt{64*7}=\sqrt{64}*\sqrt{7}=8\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2)-8\sqrt{7}}{2*1}=\frac{-2-8\sqrt{7}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2)+8\sqrt{7}}{2*1}=\frac{-2+8\sqrt{7}}{2}
$