Mathematics - Quadratic Equation Question with Solution | TestHub

${\log }_{17}{\log }_{11}(\sqrt{x+1}+\sqrt{x})=0$ Equation (1) is defined if $x\ge 0$. We can rewrite (1) as ${\log }_{11}(\sqrt{x+11}+\sqrt{x})=17^0=1$ $\Rightarrow \sqrt{x+11}+\sqrt{x}=11^1=11$ $\Rightarrow \sqrt{x+11}=11-\sqrt{x}$ Squaring both sides we get $x+11=121-22\sqrt{x}+x$ $\Rightarrow 22\sqrt{x}=110\Rightarrow \sqrt{x}=5$ or $x=25$ This clearly satisfies (1). Thus, sum of all the values satisfying (1) is 25 .