Symplfying a radical

Question

On the last substitution the odd powers of y in the radical don't seem to disappear, am I doing something wrong, please help.

Answer 1

It is most likely they meant $y = x - \frac{a+b}{2}$; note that we then have $x^2 - (a+b)x + ab = (x-a)(x-b) = (y + \frac{a+b}{2} - a)(y + \frac{a+b}{2} - b) = (y+\frac{b-a}{2})(y-\frac{b-a}{2})$ which is a difference of two squares, cancelling the odd power term. Note that the other bracket is simply this with a constant added and so no odd powers exist.