Mathematics - Quadratic Equation Question with Solution | TestHub

$a^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots \dots \dots }=a$

$$\begin{array}{rl}& \sqrt{x\sqrt{x\sqrt{x\dots \dots \infty }}}=x\text{ and }\sqrt{49+20\sqrt{6}}=5+2\sqrt{6}\\ & x^2-3>0\text{ and }x>0\\ & \Rightarrow x>\sqrt{3}\\ & (5+2\sqrt{6}{)}^{\sqrt{a\sqrt{a\sqrt{a\dots \dots }}}}+(5-2\sqrt{6}{)}^{x^2+x-3-\sqrt{x\sqrt{x\sqrt{x\dots \infty }}}}=10\\ & \Rightarrow (5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10\\ & \therefore x^2-3=1\Rightarrow x=2(\because x>\sqrt{3})\end{array}$$