|x^2-5x|=1= (x^2-5x=1) and (x^2-5x = -1)

now, you need to solve the above two quadratic equation to get the ans.

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Is the answer to blank *4*?

How do we get -1 on the RHS? Is it becuase the when we take modulus on 1 we get both -1 and +1 as the absolute values?

yes the mod opens with positive and negative sign