` sqrt ( x - a ) = x - 4 `
If ` a = 2 ` , what is the solution set of the question above ?

Options

A. ` { 3, 6} `
B. `{2} `
C. ` {3} `
D. ` {6}`

Answer

D

Explanation

The question states that ` sqrt ( x - a ) = x - 4 ` and that ` a = 2, ` so substituting 2 for a in the equation yields ` sqrt ( x - 2 ) = x - 4`. To solve for x sqaure each side of the equation , which gives ` ( sqrt ( x - 2 ) ) ^ 2 = (x -4 ) ^ 2 , or x - 2 = ( x -4 ) ^ 2 ` . then expanding ` ( x - 4) ^ 2 ` yields ` x - 2 = x ^2 -8 x + 16, or 0 = x ^ 2 - 9x + 18`. Factoring the right -hand side gives ` 0 = ( x -3 ) ( x -6 ) ` and so ` x = 3 or x = 6 ` However, for ` x = 3 ` , the original equation becomes ` sqrt ( 3 -2) = 3 -4 ` which yields ` 1 = -1 ` which is true. Hence, ` x =3 ` is an extraneous solution that arose from squaring each side of the equation. for ` x = 6`, the original equation becomes ` sqrt ( 6 - 2 ) = 6 - 4 ` which yields ` sqrt 4 = 2 , or 2 = 2 `. since this is true , the solution set of ` sqrt ( x - 2 ) = x - 4 ` is {6}.
Choice A is incorrect because it includes the extraneous solution in the solution set. Choice B is incorrect and may be the result of a calculation or factoring error. Choice C is incorrect because it includes only the extraneous solution, and not the correct solution, in the solution set.