X2+(x+1)2+(x+2)2=77

Solution for X2+(x+1)2+(x+2)2=77 equation:

X2+(X+1)2+(X+2)2=77

We move all terms to the left:

X2+(X+1)2+(X+2)2-(77)=0

We add all the numbers together, and all the variables

X^2+(X+1)2+(X+2)2-77=0

We multiply parentheses

X^2+2X+2X+2+4-77=0

We add all the numbers together, and all the variables

X^2+4X-71=0

a = 1; b = 4; c = -71;
Δ = b2-4ac
Δ = 42-4·1·(-71)
Δ = 300
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:

$\sqrt{\Delta}=\sqrt{300}=\sqrt{100*3}=\sqrt{100}*\sqrt{3}=10\sqrt{3}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-10\sqrt{3}}{2*1}=\frac{-4-10\sqrt{3}}{2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+10\sqrt{3}}{2*1}=\frac{-4+10\sqrt{3}}{2}
$