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? ax^4 +bx^3 +cx^2 +dx+e=0 1. divide by a 2. x=z−(b/(4a)) this leads to the reduced z^4 +pz^2 +qz+r=0 now we find the nature of the roots: T_1 =16p^4 r−4p^3 q^2 −128p^2 r^2 +144pq^2 r−27q^4 +256r^3 T_2 =p^2 +12r T_3 =−p^2 +4r T_1 <0 ⇒ 2 distinct real and 2 conjugated complex roots T_1 >0∧(p<0∧T_3 <0) ⇒ 4 distinct real roots T_1 >0∧(p>0∨T_3 >0) ⇒ 2 pairs of conjugated complex roots T_1 =0∧(p<0∧T_3 <0∧T_2 ≠0) ⇒ 1 real double and 2 real simple roots T_1 =0∧(T_3 >0∨(p>0∧(T_3 ≠0∨q≠0))) ⇒ 1 real double and 2 conjugated complex roots T_1 =0∧(T_2 =0∧T_3 ≠0) ⇒ 1 real triple and 1 real simple roots T_1 =0∧(T_3 =0∧p<0) ⇒ 2 real double roots T_1 =0∧(T_3 =0∧p>0∧q=0) ⇒ 2 conjugated complex double roots T_1 =0∧T


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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

$I just corrected 2 typos$ ${it}^{'} s not r but q in 2 cases . . .$
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