The sum of the solutions of the equation [| x - 2 | + x ( x - 4 ) + 2 = 0] (x 0) is equa…

Question

The sum of the solutions of the equation

$$\left| {\sqrt x  - 2} \right| + \sqrt x \left( {\sqrt x  - 4} \right) + 2 = 0$$

(x > 0) is equal to:

Accepted Answer

Case 1 : When $$\sqrt x  \ge 2$$

then $$\left| {\sqrt x  - 2} \right| = \sqrt x  - 2$$

$$ \therefore $$ The given equation becomes,

$$\left( {\sqrt x  - 2} \right)$$ + $$\sqrt x \left( {\sqrt x  - 4} \right) + 2$$ = 0

$$ \Rightarrow $$ $$\left( {\sqrt x  - 2} \right)$$ + $$x - 4\sqrt x $$ + 2 = 0

$$ \Rightarrow $$ $$x - 3\sqrt x $$ = 0

$$ \Rightarrow $$ $$\sqrt x \left( {\sqrt x  - 3} \right)$$ = 0

$$ \therefore $$ $$\sqrt x $$ = 0 or 3

$$\sqrt x $$ = 0 is not possible as $$\sqrt x  \ge 2$$.

So, $$\sqrt x $$ = 3

or $x$ = 9

Case 2 : When $$\sqrt x  < 2$$

then $$\left| {\sqrt x  - 2} \right| = $$ $ - \left( {\sqrt x  - 2} \right) ( = \]2 - \sqrt x $$ \therefore $The given equation becomes,$\left( {2 - \sqrt x } \right) ( + \]\sqrt x \left( {\sqrt x  - 4} \right) + 2$= 0$ \Rightarrow $${2 - \sqrt x }$+$x - 4\sqrt x $+ 2 = 0$ \Rightarrow $$x - 5\sqrt x  + 4$= 0$ \Rightarrow $$x - 4\sqrt x - \sqrt x  + 4$= 0$ \Rightarrow $$\sqrt x \left( {\sqrt x  - 4} \right) ( $-\left( {\sqrt x  - 4} \right)\] = 0

$$ \Rightarrow $$ $$\left( {\sqrt x  - 4} \right)$$ $\left( {\sqrt x  - 1} \right) ( = 0

\] \therefore $$\sqrt x $= 4 or 1$\sqrt x $= 4 is not possible as$\sqrt x  < 2$.$ \therefore $$\sqrt x $= 1

or$x$

Solutions · 1