# Solve[ ] with Method -> Reduce gives a different result than Reduce[ ]

Why does

Solve[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] ==
m Sqrt[x/(x + Sqrt[x])], x, Reals, Method -> Reduce]


give a different result than

Reduce[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] ==
m Sqrt[x/(x + Sqrt[x])], x, Reals]


note that the first one says $1<m<2$ while the second one says $1<m\le 2$.
The documentation says

With Method->Reduce, Solve uses only equivalent transformations and finds all solutions.

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Solve by default works with generic parameters, even if you use the option Method -> Reduce. To get the special parameter value m = 2 you need to set MaxExtraConditions to All:
Solve[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] == m Sqrt[x/(x + Sqrt[x])], x, Reals,