ЯсноПонятно24

To solve this equation, we need to consider two cases: when x is greater than or equal to a, and when x is less than a.

Case 1: x ≥ a In this case, the equation |x-a| simplifies to x-a. Substituting this into the original equation, we get: x² - x + 1/3 = x - a

Rearranging terms, we get: x² - 2x + (1/3 + a) = 0

Using the quadratic formula, we can solve for x: x = [2 ± sqrt((-2)² - 4(1)(1/3 + a))] / 2 x = [2 ± sqrt(4 - 4/3 - 4a)] / 2 x = [2 ± sqrt(12 - 4 - 12a)] / 2 x = [2 ± sqrt(-4 - 12a)] / 2

Since x ≥ a, we can discard the negative root and simplify to: x = 1 + sqrt(-4 - 12a)

Case 2: x < a In this case, the equation |x-a| simplifies to -(x-a). Substituting this into the original equation, we get: x² - x + 1/3 = -(x - a)

Rearranging terms, we get: x² + x + (1/3 + a) = 0

Using the quadratic formula, we can solve for x: x = [-1 ± sqrt(1 - 4(1)(1/3 + a))] / 2 x = [-1 ± sqrt(1 - 4/3 - 4a)] / 2 x = [-1 ± sqrt(3 - 4 - 12a)] / 2 x = [-1 ± sqrt(-1 - 12a)] / 2

Since x < a, we can discard the positive root and simplify to: x = -1 - sqrt(-1 - 12a)

Therefore, the solutions to the equation x² - x + 1/3 = |x-a| are x = 1 + sqrt(-4 - 12a) when x ≥ a, and x = -1 - sqrt(-1 - 12a) when x < a.