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Five Fives Fancy Formula (Posted on 2012-12-17) Difficulty: 3 of 5
Find all real x satisfying this equation:

√(5+√(5-√(5+√(5-√(5+x))))) = x

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution computer exploration (spoiler) | Comment 1 of 4

When graphed, the function is valid only from x=-5 to 20 as beyond this domain imaginary numbers enter. The value monotonically increases from  2.55340368400053 to 2.581183401292867 and is crossed by the line y=x only once.

FOR x = 2.5623043215017# TO 2.5623043215018# STEP .00000000000001#
  PRINT x, SQR(5 + SQR(5 - SQR(5 + SQR(5 - SQR(5 + x))))) - x

finds that the answer is approximately 2.56230432150173.

the program was changed each time the next decimal place was sought, narrowing the possibilities.

  Posted by Charlie on 2012-12-17 17:44:44
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