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Question Number 63373 by MJS last updated on 31/Jul/19

just found this on the web

I thought it might help in some cases where

quartics appear i . e . Sir {Aifour}^{'} s geometric

questions . sometimes we know the nature of

the roots, but how to use this information ?

{a x}^{4} + {b x}^{3} + {c x}^{2} + d x + e = 0

1 . divide by a

2 . x = z - \frac{b}{4 a}

this leads to the reduced

z^{4} + {p z}^{2} + q z + r = 0

now we find the nature of the roots :

T_{1} = 16 p^{4} r - 4 p^{3} q^{2} - 128 p^{2} r^{2} + 144 {p q}^{2} r - 27 q^{4} + 256 r^{3}

T_{2} = p^{2} + 12 r

T_{3} = - p^{2} + 4 r

T_{1} < 0 \Rightarrow 2 distinct real and 2 conjugated complex roots

T_{1} > 0 \land (p < 0 \land T_{3} < 0) \Rightarrow 4 distinct real roots

T_{1} > 0 \land (p > 0 \lor T_{3} > 0) \Rightarrow 2 pairs of conjugated complex roots

T_{1} = 0 \land (p < 0 \land T_{3} < 0 \land T_{2} \neq 0) \Rightarrow 1 real double and 2 real simple roots

T_{1} = 0 \land (T_{3} > 0 \lor (p > 0 \land (T_{3} \neq 0 \lor q \neq 0))) \Rightarrow 1 real double and 2 conjugated complex roots

T_{1} = 0 \land (T_{2} = 0 \land T_{3} \neq 0) \Rightarrow 1 real triple and 1 real simple roots

T_{1} = 0 \land (T_{3} = 0 \land p < 0) \Rightarrow 2 real double roots

T_{1} = 0 \land (T_{3} = 0 \land p > 0 \land q = 0) \Rightarrow 2 conjugated complex double roots

T_{1} = 0 \land T_{2} = 0 \Rightarrow all roots are equal

Commented by mr W last updated on 03/Jul/19

g o o d i n f o r m a t i o n, t h a n k s s i r!

Commented by MJS last updated on 31/Jul/19

I just corrected 2 typos

{it}^{'} s not r but q in 2 cases \dots

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