Question

How do you solve `sqrt[x-5]-sqrt[x-3]=4` and find any extraneous solutions?

Answer 1

GIven: `sqrt(x-5)-sqrt(x-3)=4`

If you multiply both sides by the conjugate, `(sqrt(x-5)+sqrt(x-3))`, the radicals on the left disappear:

`(x-5)-(x-3)=4(sqrt(x-5)+sqrt(x-3))`

Please observe that the left side simplifies to a negative number:

`-2 = 4(sqrt(x-5)+sqrt(x-3))`

Divide both sides by 4 and flip the equation:

`sqrt(x-5)+sqrt(x-3) = -1/2`

This means that no solution exists.

Think of it this way.

You are starting with

`sqrt(x-5)`

(which must be a positive number) and you are subtracting another positive number

`sqrt(x-3)`

and you obtain a positive number

4

but, if you add the two numbers, you obtain a negative number, -1/2?

No.