Question

What factors multiply to -1 but add to 1

Answer 1

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`

`(-b+- √(b^2-4ac) )/(2a) \\ \\ (1 +- √(1-4(1)(-1)) )/(2) \\ \\ (1 +- √(5) )/(2)`

x = -y =
`(1 +- √(5) )/(2)`
Either x or y could be the positive factor. The other variable would have to be the negative factor.

Just a sidenote:
`(1 +- √(5) )/(2)` is actually called the "Golden Ratio" because the ratio is found so often in beautiful example in nature.