Let the two factors be x and y.
Two factors that multiply to -1: \(xy = -1\)
Two factors that add to 1: \(x + y = 1\)
From the second equation: \(x = 1-y\)
Plug x back into the first equation: \((1-y)(y) = -1 \\ y-y^2 = -1 \\ y^2-y-1 = 0\)
There are no binomial pairs that result in this quadratic. Thus we must use the quadratic formula.
\(y^2-y-1=0\)
\(\frac{-b+- \sqrt{b^2-4ac} }{2a} \\ \\ \frac{1 +- \sqrt{1-4(1)(-1)} }{2} \\ \\ \frac{1 +- \sqrt{5} }{2}\)
x = -y = \(\frac{1 +- \sqrt{5} }{2}\)
Either x or y could be the positive factor. The other variable would have to be the negative factor.
Just a sidenote: \(\frac{1 +- \sqrt{5} }{2}\) is actually called the "Golden Ratio" because the ratio is found so often in beautiful example in nature.
