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The discriminant of the quadratic formula i.e sqrt(b^2 - 4ac) is equal to the first derivative of the original quadratic expression. I am not sure how this is derived because the author doesnt give a derivation.

Theres also one simple relation between the first derivative of an expression and the factors of it in the form (x-a)(x-b)(x-c).... which is : The sum of the factors is equal to the first derivative. This is easy to prove but i have never seen this in any other calculus book or algebra book. Heres the proof :

Lets use a quadratic expression for simplicity,y, and its factors are (x+a)(x+b)

(d/dx)y = (x+a)((d/dx)*(x+b)) + (x+b)((d/dx)*(x+a)

= x+a + x+b

Using this, you can find the factors from the derivative or viceversa.

I think the first method I described can also be derived in a similar way, but the author didnt do it in the book.