Respuesta :

-4 is an extraneous solution.

The equation is

[tex](x - 2)^{1/2}[/tex] - [tex](28-2x^{1/4}[/tex] = 0

To know whether the given equation has an extraneous solution or not we need to find the value of x.

[tex](x-2^{1/2}[/tex]= [tex](28-2x)^{1/4}[/tex]

Multiply both sides with the power of 4, and we get

[tex](x- 2)^{2}[/tex] = 28-2x

[tex]x^{2}[/tex] - 4x + 4 = 28-2x

[tex]x^{2}[/tex] - 2x - 24= 0

[tex]x^{2}[/tex]- 6x + 4x -24=0

x(x-6) +4(x-6)=0

(x-6)(x+4) = 0

x = 6 , -4

Now put both the value of x in the equation

x=6

[tex](6-2)^{1/2}[/tex]-[tex](28- 2* 6)^{1/4}[/tex]= 0

[tex]4^{1/2}[/tex]- [tex]16^{1/4}[/tex]= 0

2-2=0

0=0

Now

x = -4

[tex](-4-2)^{1/2}[/tex]-[tex](28-2*-4)^{1/4}[/tex]=0

[tex](-6)^{1/2}[/tex] -[tex]36^{1/4}[/tex]=0

[tex](-6)^{2}[/tex] - [tex]6^{1/2}[/tex]=0

Which is not possible and a negative integer is not present in the square root.

Therefore we get the extraneous solution of the equation which is x= -4, and x = 6 is the solution of the equation.

To know more about the extraneous solution refer to the link given below:

https://brainly.com/question/12361474