Quadratic formula.  

x =  (-b +/- sqrt(b^2-4ac)) / 2a

Row, row, row your boat, gently down the stream; merrily merrily merrily merrily; life is but a dream

equals [negative b], plus or minus square root; [b] squared minus four [ac] all-over two [a]

Wolfram alpha helped show me the way since I forgot about the whole completing the square thing

First I let R_{red sys} be X, and R_sys be Y, for cleanliness because my handwriting sucks. Then add 1 to each side.

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That said, I do not know why the author chose the second solution in each case.

Edit: Fixed the first image; I had accidentally cropped out the first couple of lines, which are probably kind of round-a-bout and dopey anyways...but it works out in that it allows you to complete the square, so...

BTW, I think the image in the OP has a typo; should be 3.16E-5 and not 3.6E-5. Perhaps an intentional error to catch copypasta solutions?

Looks like R is a reliability figure.  If you used the first possible solution to the quadratic equation, you'd get a reliability figure greater than 1, which is outside the possible range of answers.  Otherwise, what IS a negative probability of failure?

Note - just because a quadratic allows more than one answer, doesn't mean that all of them can actually exist in a physical system.

Mike

This.
It  is one of those places that 'Engineering Math' is used instead of purely 'academic' math.