Let $ x = {\sqrt{25 -\sqrt{51- \sqrt{25-\sqrt 51.....}}}}$ what is the value of x?
My attempt is to let $x = {\sqrt{25-\sqrt{51-x}}}$ and raised it to be in a polynomial form but I just got equation $$ x^4-50x^2+x+574=0 $$ which I couldn't solve and I ran out of idea to solve this. Thanks for help in advance!
Using the Sturm Chain for $x^4-50x^2+x+574$, it can be shown that there is one root in $[0,5]$. $$ \frac{\mathrm{d}}{\mathrm{d}x}\sqrt{25-\sqrt{51-x}}=\frac1{2\sqrt{25-\sqrt{51-x}}}\frac1{2\sqrt{51-x}}\tag1 $$ For $x\in[0,5]$, $(1)$ is bounded by $$ \frac1{4\sqrt{25-\sqrt{51}}\sqrt{46}}=\frac{\sqrt{25+\sqrt{51}}}{4\sqrt{574}\sqrt{46}}\le\sqrt{\frac{33}{422464}}\le\frac1{113}\tag2 $$ Thus, for $x\in[0,5]$, $\sqrt{25-\sqrt{51-x}}$ is a contraction. This means iterations converge to the root of $x^4-50x^2+x+574=0$ in $[0,5]$.
In fact, $10$ iterations, representing $20$ levels of square roots, yields $20$ decimal places of accuracy, since $5\cdot\left(\frac1{113}\right)^{10}=1.47294174\times10^{-20}$: $$ x=4.2618623097711113752 $$