Discriminant Of A Higher Degree Polynomials

Question

I searched a lot in Google and even tried myself to determine the nature of roots of 4th and 3rd degree standard equations.Is their any way to know nature of roots of an equation such as $(b^2-4ac)^.5$ ?

Answer 1

The quadratic, cubic and quartic discriminants can all be found here. We also get the following information about the roots from these discriminants:

$\Delta_3 > 0 \implies$ 3 distinct roots

$\Delta_3 = 0 \implies$ 2 repeated roots, all roots are real

$\Delta_3 < 0 \implies$ 1 real root, 2 conjugated complex roots

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$\Delta_4 > 0 \implies$ 4 real roots or 2 pairs of conjugated complex roots

$\Delta_4 = 0 \implies$ a minimum of 2 roots are repeated. They may be complex or real.

$\Delta_4 < 0 \implies$ a pair of conjugated complex roots and 2 real roots. All of these are distinct.

The source for the above conditions can be found here. Note that it may also be easier to in practice work with the reduced cubic (repectively reduced quartic) equations, the formulae/substitutions for which can easily be found online.