We can start by simplifying the left side of the equation:

√(X^2 + x + 1/4) = √[(X + 1/2)^2]

Using the identity √(a^2) = |a|, we can simplify further:

√[(X + 1/2)^2] = |X + 1/2|

Now we can rewrite the original equation as:

|X + 1/2| = x - X

There are two cases to consider:

Case 1: X + 1/2 ≥ 0

In this case, we can simplify the equation to:

X + 1/2 = x - X

Solving for X, we get:

X = (x - 1/4)/2

Note that this solution is valid only if x ≥ 1/4.

Case 2: X + 1/2 < 0

In this case, we can simplify the equation to:

-(X + 1/2) = x - X

Solving for X, we get:

X = (x + 1/2)/2

Note that this solution is valid only if x ≤ -1/2.

Therefore, the solution to the equation is:

X = (x - 1/4)/2, if x ≥ 1/4

X = (x + 1/2)/2, if x ≤ -1/2

Note that if -1/2 < x < 1/4, there is no real solution to the equation.