Mathematics - Quadratic Equation Question with Solution | TestHub

Let $x=\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}}}}\dots \dots$ $\Rightarrow x=\frac{1}{3\sqrt{2}}\sqrt{4-x}$ $\Rightarrow 3\sqrt{2}x=\sqrt{4-x}$ $\begin{array}{l}\Rightarrow 18x^2=4-x\\ \Rightarrow 18x^2+x-4=0\\ \Rightarrow (9x-4)(2x+1)=0\\ \Rightarrow x=\frac{4}{9}\text{ or }x=-\frac{1}{2}\end{array}$ (Not possible because log is not define for -ve value) $\begin{array}{rl}\therefore & 6+{\log }_{\frac{3}{2}}\left(\frac{4}{9}\right)=6+{\log }_{\frac{3}{2}}{\left(\frac{3}{2}\right)}^{-2}\\ & =6-2=4\end{array}$