how to solve infinite root 5 problems | infinite square root of 5| how to solve infinite nested radicals

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how to solve infinite root 5 problems and solutions

x=\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-...}}}}}

 Squaring both sides, we get 

x^2=5-\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-...}}}}}
x^2=5-x\\ \\ x^2+x-5=0

solve this using the quadratic formula.

a=1\; , b=1 ,\;c=-5
x=\frac{-b\;\pm \sqrt{b^2-4ac}}{2a}
x=\frac{(-1)\;\pm \sqrt{1-4(-5)}}{2}
=\frac{-1\pm \sqrt{21}}{2}

Remember, positive square roots x is a square root and thus a positive square root

so x\geq 0

only\;x=\frac{-1+ \sqrt{21}}{2} \;\;is \;valid \;answer

keywords

root 5 root 5 root 5 upto infinity


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