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Q.
Sum of all the values of $x$ satisfying the equation
$\log _{17} \log _{11}(\sqrt{x+11}+\sqrt{x})=0$
is:
Complex Numbers and Quadratic Equations
Solution:
Equation (1) is defined if $x \geq 0$.
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 the sides we get
$x+11 =121-22 \sqrt{x}+x$
$\Rightarrow 22 \sqrt{x} =110 \Rightarrow \sqrt{x}=5 \text { or } x=25$
This clearly satisfies (1).
Thus, sum of all the values satisfying (1) is 25 .