Re: Equation de géométrie impossible pour Maxima et d'autres

Le mardi 28 novembre 2023 à 11:14:17 UTC+1, Michel Talon a écrit :

Le 28/11/2023 à 10:24, efji a écrit :

L'équation me semble du type
 
a*sqrt(P(x)) + b*sqrt(Q(x)) + c = 0
 
avec P et Q des polynômes de degré 2.
On ne peut pas s'en sortir analytiquement pour éliminer les racines (il
me semble),

Si on peut, maxima:
(%i1) eq1:a*sqrt(P(x))=c-b*sqrt(Q(x));
(%o1) a*sqrt(P(x)) = c-b*sqrt(Q(x))
(%i2) expand(eq1^2);
(%o2) a^2*P(x) = b^2*Q(x)-2*b*c*sqrt(Q(x))+c^2
(%i3) eq2:b^2*Q(x)+c^2-a^2*P(x)=2*b*c*sqrt(Q(x));
(%o3) b^2*Q(x)-a^2*P(x)+c^2 = 2*b*c*sqrt(Q(x))
(%i4) expand(eq2^2);
(%o4)
b^4*Q(x)^2-2*a^2*b^2*P(x)*Q(x)+2*b^2*c^2*Q(x)+a^4*P(x)^2-2*a^2*c^2*P(x)
+c ^4 = 4*b^2*c^2*Q(x)
 
qui est de degré 4 en x via Q^2 et P*Q donc soluble explicitement dans
maxima.
Cela étant avec une formule monstrueuse.
 
--
Michel Talon

Merci de la réponse
J'y ai pensé après avoir posté mais Maxima me donne des solutions qui sont fausses sur le tableur
Je divise par a pour passer de aX^4 à X^4
et  avoir directement les cases libre office remplies
en posant
A1 B1 C1  D1 et E1
je calcule
F1=B1/A1
G1 = C1/A1
H1 = D1/A1
I1= E1/A1
je demande à Maxima les 4 racines

(%i1) S^4+F1*S^3+G1*S^2+H1*S+I1;

Maxima me donne


S1=-racine(-(3*(-8*H1+4*F1*G1-F1^3))/(2*racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)))-(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+((H1*F1-4*I1)/3+((-1)*G1^2)/9)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)-(4*G1)/3+F1^2/2)/2-racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3))/12+((-1)*F1)/4

S2=racine(-(3*(-8*H1+4*F1*G1-F1^3))/(2*racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)))-(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+((H1*F1-4*I1)/3+((-1)*G1^2)/9)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)-(4*G1)/3+F1^2/2)/2-racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3))/12+((-1)*F1)/4

S3=-racine((3*(-8*H1+4*F1*G1-F1^3))/(2*racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)))-(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+((H1*F1-4*I1)/3+((-1)*G1^2)/9)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)-(4*G1)/3+F1^2/2)/2+racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3))/12+((-1)*F1)/4

S4=racine((3*(-8*H1+4*F1*G1-F1^3))/(2*racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)))-(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+((H1*F1-4*I1)/3+((-1)*G1^2)/9)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)-(4*G1)/3+F1^2/2)/2+racine((36*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(2/3)+(9*F1^2-24*G1)*(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3)+48*I1-12*F1*H1+4*G1^2)/(racine(-256*I1^3-(-192*F1*H1-128*G1^2+144*F1^2*G1-27*F1^4)*I1^2-((144*G1-6*F1^2)*H1^2+(18*F1^3*G1-80*F1*G1^2)*H1+16*G1^4-4*F1^2*G1^3)*I1+27*H1^4-(18*F1*G1-4*F1^3)*H1^3-(F1^2*G1^2-4*G1^3)*H1^2)/(2*3^(3/2))+(-G1*(H1*F1-4*I1)-3*(-I1*F1^2+4*I1*G1-H1^2))/6+G1^3/27)^(1/3))/12+((-1)*F1)/4