Bonsoir,
A(x) = 9 - (2x - 5)^2
Developper :
A(x) = 9 - (4x^2 - 20x + 25)
A(x) = 9 - 4x^2 + 20x - 25
A(x) = -4x^2 + 20x - 16
Factoriser :
A(x) = 9 - (2x - 5)^2
A(x) = 3^2 - (2x - 5)^2
A(x) = (3 - 2x + 5)(3 + 2x - 5)
A(x) = (-2x + 8)(2x - 2)
A(x) = 4(-x + 4)(x - 1)
Calculer A(0) :
A(0) = 4(-0 + 4)(0 - 1)
A(0) = 4 * 4 * (-1)
A(0) = -16
Résoudre A(x) = 0 :
4(-x + 4)(x - 1) = 0
Pour qu’un produit soit nul il faut qu’au moins un de ses facteurs soit nul :
-x + 4 = 0
x = 4
Ou
x - 1 = 0
x = 1
S = {1;4}
Résoudre :
2x + 4 = -3x + 7
2x + 3x = 7 - 4
5x = 3
x = 3/5
(1 - 2x)(6x - 5) = 0
1 - 2x = 0
2x = 1
x = 1/2
Ou
6x - 5 = 0
6x = 5
x = 5/6
(4x - 1)/(x + 1) = 0
4x - 1 = 0
4x = 1
x = 1/4
Avec x + 1 # 0
x # -1
Résoudre :
3x - 3 < 1 - 2x
3x + 2x < 1 + 3
5x < 4
x < 4/5
x € ]-inf;4/5[
(x - 2)/3 - (1-x)/2 << 0
(2x - 4)/6 - (3-3x)/6 << 0
2x - 4 - 3 + 3x << 0
5x << 7
x << 7/5
x € ]-inf ;7/5]
(2x - 3)(1 - 7x) < 0
2x - 3 = 0
2x = 3
x = 3/2
Ou
1 - 7x = 0
1 = 7x
x = 1/7
x.........|.-inf............1/7.............3/2...........+inf
2x-3..|..........(-)..............(-)........||......(+)..........
1-7x...|...........(+).....||......(-)................(-)..........
Eq.....|...........(-)......||......(+)......||.......(-)..........
x € ]-inf;1/7[ U ]3/2;+inf[
x(x + 1)/(3 - 2x) << 0
x = 0
Ou
x + 1 = 0
x = -1
Ou
3 - 2x = 0
2x = 3
x = 3/2
x........|....-inf..........(-1)............0..........3/2......+inf
x........|...........(-)..............(-).....O....(+)........(+).......
x+1....|...........(-).......O.....(+)............(+).......(+)......
3-2x..|...........(+).............(+)...........(+)....O....(-).....
Eq.....|...........(+)......O.....(-)....O...(+)....O....(-)......
x € [-1;0] U [3/2;+inf[
