How to extract expression from ConditionalExpression

You can use Normal, ConditionalExpression is not explicitly mentioned there but documentation says it deals with special forms.

p1 = y /. {First[Solve[x^2 + y^2 + x == 1, y, Reals]]} // First

 ConditionalExpression[-Sqrt[1 - x - x^2], 1/2 (-1 - Sqrt[5]) < x < 1/2 (-1 + Sqrt[5])]

Normal @ p1

-Sqrt[1 - x - x^2]

You can forcely specify the condition to be True:

Solve[x^2 + y^2 + x == 1, y, Reals] /.
 ConditionalExpression[e_, _] :> ConditionalExpression[e, True]

{{y -> -Sqrt[1 - x - x^2]}, {y -> Sqrt[1 - x - x^2]}}

But you should always keep it in mind that this is not an identical transformation.

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