Question

The sum of the solutions of the equation |√x - 2| + √x(√x - 4) + 2 = 0, (x > 0) is __ ?
(Is it 10?)

Answer 1

Answer :

10.

Solution :

$\tt \: Q. \: \: f(x) = | \sqrt{x} - 2 | + \sqrt{x} ( \sqrt{x} - 4) + 2 = 0, \: where \: x>0.$

Now, Let √x = t.

=> t - 2 = 0.

=> t = 2.

______________

Case 1.

• t ≥ 2.

$\tt \longrightarrow t - 2 + {t}^{2} - 4t + 2 = 0.$

$\tt \longrightarrow {t}^{2} -3t = 0.$

$\tt \longrightarrow t= 3.$

$\tt \longrightarrow \sqrt{x} = 3.$

$\tt \longrightarrow x = 9 \: , \: 0.$

______________

Case 2.

• x < 2.

$\tt \longrightarrow 2 - t + {t}^{2} - 4t + 2 = 0.$

$\tt \longrightarrow {t}^{2} - 5t + 4 = 0.$

$\tt \longrightarrow (t - 4)(t - 1)= 0.$

$\tt \longrightarrow t= 1 \: , 4 .$

$\tt \longrightarrow \sqrt{x} = 1 \: , 4.$

$\tt \longrightarrow x = 1 \: , 16.$

___________________________

Substitute the solution of given equation

$\tt \longrightarrow| \sqrt{16} - 2 | + \sqrt{16} ( \sqrt{16} - 4) + 2 = 0.$

$\tt \longrightarrow| 4 - 2 | + 4 (4 - 4) + 2 = 0.$

$\tt \longrightarrow| 2 | + 0 + 2 = 0.$

$\tt \longrightarrow4 \neq 0.$

___________________________

$\tt \longrightarrow \red{x \neq 16 \: , \: 0}.$

• Sum of all solution of given equation is 1 + 9 = 10.
• The sum of the solutions of the function f( x ) is 10.

Answer 2

$\large \dag$ Question :-

The sum of the solutions of the equation

$| \sqrt{x} - 2 | + \sqrt{x}( \sqrt{x} - 4) + 2 = 0 \: \:where \: \: (x > 0)$

Is, ____?.

$\large \dag$ Answer :-

$\sf\tt\large{\red {\underline {\underline{⚘\;The \;Sum \;of \;the \;Solutions \;the \;equation \;is:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;10 :}}}}$

$\large \dag$ Solutions :-

According to the given question,

Consider,

$\sqrt{x} = k$

Where,

• K - 2 = 0
• k = 2.

Here,

First consider two case ,

Now, lets take

$\large \dag$ (i) Case :-

$k \geqslant 2$

$k - 2 + {k}^{2} - 4k + 2 = 0$

${k}^{2} - 3k = 0 = k = 3$

$\sqrt{x} = 3$

$x = 9,0$

Now,

$\large \dag$ (ii)Case :-

$k < 2$

$2 - k + {k}^{2} - 4k + 2 = 0$

${k}^{2} - 5k + 4 = 0$

$(k - 4)(k - 1) = 1,4$

$\sqrt{x} = 1,4$

$x = (1,16)$

Now,

• Here by substituting all of the values we get that ,

$| \sqrt{16} - 2| + \sqrt{16} ( \sqrt{16} - 4) + 2 = 0$

$= |4 - 2| + 4(4 - 4) + 2 = 0$

$|2| + 0 + 2 = 0$

Therefore,

• 4 is not equal to zero .
• And also we can tell that,

• x is not equal to 16,0

At finally we come to know that,

• The sum of solutions of the equation of the given question is 10.

$\large \dag$ Hope it helps u mate .

$\large \dag$ Thank you .