Answer by mvw for There is only one distinct pair of numbers that multiply to a given number and sum to a given number.

You start with$$ x + y = c_1 \\x \, y = c_2$$and transform to$$y = c_1 - x \\x (c_1 - x) = c_2$$where the second equation can be transformed to$$0 = x^2 - c_1 x + c_2 = (x - c_1/2)^2 + c_2 - c_1^2/4 \iff \\x = \frac{c_1 \pm \sqrt{c_1^2 - 4 c_2}}{2}$$so depending on $\Delta = c_1^2 - 4 c_2$ we have zero, one or two solutions $(x, y)$.