Question

$\sqrt{20 + \sqrt{20 - \sqrt{20 + \sqrt{20 - \sqrt{20 + ......... \infty } } } } }$

Answer 1

$\sqrt{20+\sqrt{20-\sqrt{20+\sqrt{20-\sqrt{20+......\infty}}}}}=x (let )\\\\\sqrt{20+\sqrt{20-x}}=x \\\\take\:square\:both\:sides,\\\\20+\sqrt{20-x}=x^2\\\\(x^2-20)=\sqrt{20-x}\\\\take\:square\: gain,\\\\(x^2-20)^2=20-x\\\\x^4-40x^2+400-20+x=0\\\\x^4-4x^3+4x^3-16x^2-24x^2+96x-95x+380=0\\\\(x-4)(x^3+4x^2-24x-95)=0\\\\(x-4)[x^3+5x^2-x^2-5x-19x-95]=0\\\\(x-4)(x+5)(x^2-x-19)=0\\\\x=4,-5\: and\: x^2-x-19 = 0$

hence, x = 4 ,-5 , (1 +sqrt{77})/2 , (1-sqrt{77})/2
but x can't be negative so, x = 4 and (1+sqrt{77})/2  are answer