Question

Which statement about sqrt x-5 minus sqrt x=5 true?

-3 is a true solution

-3 is a extraneous solution

9 is a true solution

9 is a extraneous solution

Answer 1

D.) x = 9 is an extraneous solution.

Explanation:

Answer 2

9 is an extraneous solution.

Explanation:

Extraneous solution is a solution that when plugged into the equation do not hold true.

Here we are given an algebraic equation in terms of single variable x as:

`√(x-5)-√(x)=5-----(1)`

Now, we will solve this equation to obtain the solution as:

on squaring both side of the equation we obtain:

`(√(x-5)-√(x))^2=5^2`

`x-5+x-2√(x-5)√(x)=25\\\\2x-5-2√(x-5)√(x)=25\\\\2x-30=2√(x-5)√(x)\\\\on\ dividing\ both\ side\ by\ 2\ we\ obtain:\\\\x-15=√(x-5)√(x)`

Again on squaring both side of the equation we obtain:

`x^2+225-30x=(x-5)x\\\\\x^2+225-30x=x^2-5x\\\\225=-5x+30x\\\\225=25x\\\\x=9`

Now when we plug x=9 back to the original equation i.e. equation (1) we get:

`√(9-5)-√(9)=5\\\\√(4)-√(9)=5\\\\2-3=5\\\\-1=5`

Hence, the equation does not hold true.

Hence, 9 is a extraneous solution