Square both sides of the equation.
x^2 + 2x + a = -(1/2) + sqroot(1/4 - a)
x^2 +2x = -a -1/2 +sqrt(1/4 -a)
x^2 +2x +1 = 1 -a -1/2 +sqrt(1/4 -a)
(x +1)^2 = 1/2 -a +sqrt(1/4 -a)
I am not successful yet on how to make the RHS a perfect square too.
So, for now,
x +1 = +,-sqrt[1/2 -a +sqrt(1/4 -a)]
x = -1 +,-sqrt[1/2 -a +sqrt(1/4 -a)] ------------answer.