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And now, substituting into the formula for \(r\) and \(s\), we get \(r, s = -\frac{b}{2} \pm \sqrt{\frac{b^2}{4} - c}\).

But this is just the same as \(\frac{-b \pm \sqrt{b^2 - 4c}}{2}\), which is exactly what we would expect from the standard formula, given that \(a=1\)! The general derivation, for \(a \neq 1\), takes a few more steps, but is fairly straightforward.

Siva Kalyan@skalyan@mathstodon.xyzThe innovation here seems to be in the emphasis placed on the properties of the sum of the roots and the product of the roots.