To solve the problem of finding the value of the infinite nested square root ( \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}} ):

Step 1: Define the variable

Let ( x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}} ). Since the nested sequence is infinite, the part inside the first square root is the same as ( x ) itself.

Step 2: Square both sides

Squaring both sides of the equation gives:
( x^2 = 2 + x )

Step 3: Solve the quadratic equation

Rearrange into standard quadratic form:
( x^2 - x - 2 = 0 )

Factor the equation:
( (x - 2)(x + 1) = 0 )

Step 4: Select the positive solution

Since ( x ) is a square root (non-negative), we discard the negative root ( x = -1 ). Thus:
( x = 2 )

Answer: (\boxed{2})

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