Magnetism as a Relativistic Phenomenon

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Consider a wire with charges moving with speed 'v'and '-v' in frame F (with charge q at distance s away moving at speed u).

In frame F', the charge q is stationary, and so the speeds of the charges in the wire are v+ and v-, where v- > v+

Hi guys, I have trouble following through the proof in step 12.81 and 12.82:

How did they get rid of the ± in the denominator? I tried using a2 - b2 = (a+b)(a-b) but it doesn't match the expression:

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The plus and minus on top of each other are different solutions to the equation. Whether you choose to solve for positive or negative gamma in equation 12.81, the denominator comes out the same. Then, if you factor the denominator, you get the equation after 12.81.
 
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