(х-6)^2+2 |х-6|-24=0

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To solve this equation, we first need to separate it into two cases based on the absolute value:

Case 1: x - 6 ≥ 0
In this case, the absolute value is simply |x - 6| = x - 6. The equation becomes:
(x - 6)^2 + 2(x - 6) - 24 = 0

Expanding and simplifying the equation:
x^2 - 12x + 36 + 2x - 12 - 24 = 0
x^2 - 10x = 0
x(x - 10) = 0

Therefore, x = 0 or x = 10. But since we are in Case 1 where x ≥ 6, the solution is x = 10.

Case 2: x - 6 < 0
In this case, the absolute value is |-x + 6| = x - 6. The equation becomes:
(x - 6)^2 - 2(x - 6) - 24 = 0

Expanding and simplifying the equation:
x^2 - 12x + 36 - 2x + 12 - 24 = 0
x^2 - 14x + 24 = 0

We can solve this quadratic equation using the quadratic formula:
x = [14 ± sqrt((14)^2 - 4124)] / 2
x = [14 ± sqrt(196 - 96)] / 2
x = [14 ± sqrt(100)] / 2
x = [14 ± 10] / 2
x = 12 or x = 2

But since we are in Case 2 where x < 6, the solution is x = 2.

Therefore, the solutions to the original equation are x = 2 and x = 10.