Petit coup de main pour réussir - Clg Charles Doche

Petit coup de main pour réussir
les développements avec les identités remarquables.(Corrigé)

1- Carré d'une somme :
(a + b)2 = a2 + 2 x a x b
+ b2

A = (x + 3)2 ( a = x et b = 3 )
A = x2 + 2 x x x 3 + 32 (On applique la formule)
A = x2 + 6 x + 9

B = (3x + 5)2 ( a = 3x et b = 5 )
B = (3x)2 + 2 x 3x x 5 + 52 (Attention ! : a2 = (3x)2)
B = 9 x2 + 30 x + 25 ( (3x)2 = 3x x 3x = 9 x2 )

Exercice : Développer à l'aide de l'identité remarquable précédente :

C = (x + 6)2 = x2 + 12 x + 36

D = (4 + x)2 = 16 + 8 x + x2 = x2 + 8 x + 16

E = (2x + 7)2 = (2x)2 + 2 x 2x x 7 + 72 = 4 x2 + 28 x + 49

F = (5x + 9)2 = (5x)2 + 2 x 5x x 9 + 92 = 25 x2 + 90 x + 81

G = (8 + 3x)2 = 82 + 2 x 8 x 3x + (3x)2 = 64 + 48 x + 9 x2 = 9 x2 + 48 x +
64

H = (1 + 0,5x)2 = 12 + 2 x 1 x 0,5x + (0,5x)2 = 0,25 x2 + x + 1


2- Carré d'une différence :
(a - b)2 = a2 - 2 x a x b
+ b2

A = (x - 5)2 ( a = x et b = 5 )
A = x2 - 2 x x x 5 + 52 (On applique la formule)
A = x2 - 10 x + 25

B = (2x - 3)2 ( a = 2x et b = 3 )
B = (2x)2 - 2 x 2x x 3 + 32 (Attention ! : a2 = (2x)2 )
B = 4 x2 - 12 x + 9 ( (2x)2 = 2x x 2x = 4x2 )

Exercice : Développer à l'aide de l'identité remarquable précédente:

C = (x - 4)2 = x2 - 8 x + 16

D = (7 - x)2 = 49 - 14 x + x2 = x2 - 14 x + 49

E = (3x - 8)2 = (3x)2 - 2 x 3x x 8 + 82 = 9 x2 - 48 x + 64
F = (6x - 5)2 = (6x)2 - 2 x 6x x 5 + 52 = 36 x2 - 60 x + 25

G = (1 - 9x)2 = 12 - 2 x 1 x 9x + (9x)2 = 1 - 18 x + 81 x2 = 81 x2 - 18 x +
1

H = (12 - 10x)2 = 122 - 2 x 12 x 10x + (10x)2 = 100 x2 - 240 x + 144


3- Somme de deux carrés :


A = (x + 7)2 + (x + 5)2
A = [1 x2 + 2 x 7 x x + 72] + [1 x2 + 2 x 5 x x + 52]
A = [1 x2 + 14 x + 49] + [1 x2 + 10 x + 25]
A = 1 x2 + 14 x + 49 + 1 x2 + 10 x + 25
A = 2 x2 + 24 x + 74

B = (4x + 3)2 + (9x + 2)2
B = [(4 x)2 + 2 x 3 x 4x + 32] + [(9 x)2 + 2 x 2 x 9x + 22]
B = [16 x2 + 48 x + 9] + [81 x2 + 36 x + 4]
B = 16 x2 + 48 x + 9 + 81 x2 + 36 x + 4
B = 97 x2 + 84 x + 13

C = (x - 3)2 + (x - 10)2
C = [x2 - 6 x + 9] + [x2 - 20 x + 100]
C = x2 - 6 x + 9 + x2 - 20 x + 100
C = 2 x2 - 26 x + 109

D = (8x - 7)2 + (5x - 6)2
D =[64x2 - 112 x + 49] + [25x2 - 60 x + 36]
D = 64 x2 - 112 x + 49 + 25 x2 - 60 x + 36
D = 89 x2 - 172 x + 85

E = (2x + 5)2 + (7x - 4)2
E =[4x2 + 20 x + 25] + [49x2 - 56 x + 16]
E = 4 x2 + 20 x + 25 + 49 x2 - 56 x + 16
E = 53 x2 - 36 x + 41

F = (2x - 8)2 + (9x + 1)2
F =[4x2 - 32 x + 64] + [81x2 + 18 x + 1]
F = 4 x2 - 32 x + 64 + 81 x2 + 18 x + 1
F = 85 x2 - 14 x + 65






4- Différence de deux carrés :

A = (x + 4)2 - (x + 3)2
A = [1 x2 + 2 x 4 x x + 42] - [1 x2 + 2x 3 x x + 32]
A = [1 x2 + 8 x + 16] - [1 x2 + 6 x + 9]
A = 1 x2 + 8 x + 16 - 1 x2 - 6 x - 9 ( « - » devant le crochet ! )
A = 2 x + 7

B = (3x + 4)2 - (2x + 5)2
B = [(3 x)2 + 2 x 4 x 3x + 42] - [(2 x)2 + 2x 5 x 2x + 52]
B = [9 x2 + 24 x + 16] - [4 x2 + 20 x + 25]
B = 9 x2 + 24 x + 16 - 4 x2 - 20 x - 25 ( « - » devant le crochet !
)
B = 5 x2 + 4 x - 9

C = (x - 9)2 - (x - 8)2
C = [ x2 - 18 x + 81] - [ x2 - 16 x + 64]
C = x2 - 18 x + 81 - x2 + 16 x - 64
C = - 2 x + 17

D = (5x - 10)2 - (8x - 7)2
D = [ 25 x2 - 100 x + 100] - [ 64 x2 - 112 x + 49]
D = 25 x2 - 100 x + 100 - 64 x2 + 112 x - 49
D = - 39 x2 + 12 x + 51

E = (2x - 5)2 - (3x + 4)2
E = [ 4 x2 - 20 x + 25] - [ 9 x2 + 24 x + 16]
E = 4 x2 - 20 x + 25 - 9 x2 - 24 x - 16
E = - 5 x2 - 44 x + 9

F = (x - 6)2 - (5x + 1)2
F = [ x2 - 12 x + 36] - [ 25 x2 + 10 x + 1]
F = x2 - 12 x + 36 - 25 x2 - 10 x - 1
F = - 24 x2 - 22 x + 35

G = (4x + 6)2 + (2x - 5)2 - (7x - 6)2
G = [ 16 x2 - 48 x + 36 ] + [ 4 x2 - 20 x + 25 ] - [ 49 x2 - 84 x + 36 ]
G = 16 x2 - 48 x + 36 + 4 x2 - 20 x + 25 - 49 x2 + 84 x - 36
G = 16 x2 + 4 x2 - 49 x2 - 48 x - 20 x + 84 x + 36 +25 - 36
G = -29 x2 + 16 x + 25