SOLUTION: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?

Algebra ->  Radicals -> SOLUTION: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?      Log On


   

Question 1134096: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
x=sqrt%282%2Bsqrt%282%2Bsqrt%282%2Bsqrt%282%2B%22...%22%29%29%29%29+

Square both sides:

x%5E2=2%2Bsqrt%282%2Bsqrt%282%2Bsqrt%282%2Bsqrt%282%2B%22...%22%29%29%29%29

Subtract 2 from both sides:

x%5E2-2=sqrt%282%2Bsqrt%282%2Bsqrt%282%2Bsqrt%282%2B%22...%22%29%29%29%29+

The right side is the same as the right side of the
original equation so it's just x:

x%5E2-2=x

x%5E2-x-2=0

x%2B1%29%28x-2%29=0

x+1=0;  x-2=0
  x=-1;   x=2

We discard the negative answer because the radical
always means the POSITIVE square root.

Answer: x=2

Edwin