A2853** – La formule surprise
Déterminer le produit des solutions réelles de l’équation :
Proposition de Marc Humery
A(x) = x²-7x+11 ; B(x) = x²-13x+42 => A(x)B(x) = 1 3 possibilités :
A(x)B(x) = (-1)2m = 1 => A(x) = -1 et B(x) = 2m B(x).Ln A(x) = Ln 1 = 0 => A(x) = 1 ou B(x) = 0
1/ A(x) = -1 et B(x) = 2m (pair)
A(x) = x²-7x+11 = -1 x²-7x+12 = (x-3)(x-4) = 0 => x = 3 ; x = 4 x = 3 => A(3) = -1 ; B(3) = 12 pair => (-1)12 = 1
x = 4 => A(4) = -1 ; B(4) = 6 pair => (-1)6 = 1
2/ A(x) = 1
A(x) = x²-7x+11 = 1 x²-7x+10 = (x-2)(x-5) = 0 => x = 2 ; x = 5 x = 2 => A(2) = 1 ; B(2) = 20 => (1)20 = 1
x = 5 => A(5) = 1 ; B(5) = 2 => (1)2 = 1
3/ B(x) = 0 ; A(x) > 0
B(x) = x²-13x+42 = (x-6)(x-7) = 0 => x = 6 ; x = 7 x = 6 => B(6) = 0 ; A(6) = 5 > 0 => 50 = 1
x = 7 => B(7) = 0 ; A(7) = 11 > 0 => 110 = 1
4/ produit P des solutions réelles de l’équation A(x)B(x) = 1 P = 2.3.4.5.6.7 = 7 ! = 5 040