I don't know how he went from there to here

That's easy.

im(j) = j
Therefore, 2abj = j, so 2ab = 1 (cancel the 'j's), and either a = b or a = -b, so:
2a² = 1 or -2a² = 1
So a = sqrt(1/2) or a = sqrt(1/2) * j

That makes the roots: sqrt(1/2) + sqrt(1/2)j (a = b)
sqrt(1/2) * j - sqrt(1/2) * j * j (a = -b)

Working through the a = -b case, we get:

sqrt(1/2) * j - sqrt(1/2) * -1
Which gives:
sqrt(1/2) + sqrt(1/2) * j

So there is only one root.

Your mistake is that you left out the second j on the b term.

I don't know what you did to check it. Specifically, you seem to have put the 'j' back in between these two lines:

root(j) = root(-1/2) - root(-1/2)j, or, root(j) = - root(-1/2) + root(-1/2)j
root(j) = j/root(2) + j^2/root(2)

Which is why the check worked.