Viete's Formulas - what are they used for?

If we have a quadratic equation 2X^2 + 4X - 6 = 0, whose roots are x1=1 and x2=-3 and where a = 2 b=4 and c=-6 then according to Viete's formulas (x1 + x2) = (-b/a) and (x1 • x2) = (c/a).

I suppose these formulas could be used as a double check for calculating the roots but do they serve any other purpose? (I also know there are other Viete's formulas for cubic, quartic, etc. equations).

For quadratics, these formulas are used mainly for factoring.

Your equation can be written as ...

... 2 (x² + 2x - 3) = 0

You factor this by looking for factors of - 3 (c=x1·x2) that add to give + 2 (b = - (x1+x2)). These are {-1, + 3}, so the factorization is ...

... 2 (x - 1) (x + 3) = 0

The roots are then 1 and - 3, which sum to - b = - 2.

(You will note that the numbers used in the binomial factors are the opposites of the roots x1 and x2 in the Viete's Formulas. That is how we can look for them to sum to "b", rather than "-b".)