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To solve this equation, we can first rewrite it as:

x² + x + 1/3 = |x - a|

Next, we can consider the absolute value function and its two cases:

Case 1: x - a ≥ 0 In this case, the absolute value function becomes x - a. So the equation becomes: x² + x + 1/3 = x - a

Simplify the equation: x² + x - x + 1/3 + a = 0 x² + 1/3 + a = 0

Case 2: x - a < 0 In this case, the absolute value function becomes -(x - a) = a - x. So the equation becomes: x² + x + 1/3 = a - x

Simplify the equation: x² + x + x + 1/3 - a = 0 x² + 2x + 1/3 - a = 0

Now, we have two separate equations to solve for x. We can solve each equation separately using the quadratic formula:

For the first case: x² + 1/3 + a = 0 x = (-1 ± √(1 - 4(1/3)(a)))/2 x = (-1 ± √(1 - 4/3a))/2

For the second case: x² + 2x + 1/3 - a = 0 x = (-2 ± √(4 - 4(1/3)(1/3 - a)))/2 x = (-2 ± √(4 - 4/3 + 4a))/2 x = (-2 ± √(12a - 11))/2

Therefore, the solutions for x are: x = (-1 ± √(1 - 4/3a))/2 x = (-2 ± √(12a - 11))/2

These are the solutions to the equation x² + x + 1/3 = |x - a|, considering the two cases of the absolute value function.