MHB Solving an Equation to find the value of x

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Calculate value of $x$ in $\displaystyle \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}} = x$
 
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Hint: let $\displaystyle x+1=x^2$
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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