To solve the problem of finding the value of the infinite nested radical ( x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}} ):

Step 1: Set up the equation

Since the expression under the square root repeats infinitely, the part inside the first square root is ( 2 + x ). Thus:
[ x = \sqrt{2 + x} ]

Step 2: Square both sides

[ x^2 = 2 + x ]

Step 3: Rearrange into a quadratic equation

[ x^2 - x - 2 = 0 ]

Step 4: Solve the quadratic equation

Using the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) where ( a=1, b=-1, c=-2 ):
[ x = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} ]

Since ( x ) is positive, we take the positive root:
[ x = \frac{1 + 3}{2} = 2 ]

Answer: (\boxed{2})

作者声明:本文包含人工智能生成内容。