Mathematics - Quadratic Equation Question with Solution | TestHub

We have,$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$

$\sqrt{(\sqrt{x-1}{)}^2-2\times 2\sqrt{x-1}+4}+\sqrt{(\sqrt{x-1}{)}^2-2\times 3\sqrt{x-1}+9}=1$

$\Rightarrow \sqrt{(\sqrt{x-1}-2{)}^2}+\sqrt{(\sqrt{x-1}-3{)}^2}=1$

$\Rightarrow |\sqrt{x-1}-2|+|\sqrt{x-1}-3|=1$

$\Rightarrow |\sqrt{x-1}-2|+|\sqrt{x}-1-3|=\left|(\sqrt{x-1}-2)-(\sqrt{x-1}-3)\right|$

We know that, 

If $|x-a|+|x-b|=\left|(x-a)-(x-b)\right|$

then $(x-a)(x-b)<0$

$\therefore (\sqrt{x-1}-2)(\sqrt{x-1}-3)<0$

$\Rightarrow 2<\sqrt{x}-1<3\Rightarrow 5<x<10$

$\therefore$ Equation has infinite many solutions.