Question

How do you solve and check for extraneous solutions in `sqrt(x+3) =x-3`?

Answer 1

Square both sides, solve the resulting quadratic equation and discard the extraneous solution to find: `x=6`

Explanation:

First note that we require `x >= 3` since the left hand side is non-negative.

Now square both sides to get:
`x+3 = x^2-6x+9`

Subtract `x+3` from both sides to get:

`0 = x^2-7x+6 = (x-1)(x-6)`

This has two solutions: `x=1` and `x=6`, but `x=1` is not a solution of the original equation since we require `x >= 3`.

So the only solution to the original equation is `x=6`.