Réponse :
a^2 - b^2 = (a - b)(a + b)
a^2 - 2ab + b^2 = (a - b)^2
a^2 + 2ab + b^2 = (a + b)^2
Un produit de facteur est nul si et seulement si au moins un de ses facteurs est nul
Explications étape par étape
Bonjour
Résoudre en ramenant à une équation produit nul :
Exercice 73 :
x^2 - 36 = 0
x^2 - 6^2 = 0
(x - 6)(x + 6) = 0
x - 6 = 0 ou x + 6 = 0
x = 6 ou x = -6
4x^2 - 1 = 0
(2x)^2 - 1^2 = 0
(2x - 1)(2x + 1) = 0
2x - 1 = 0 ou 2x + 1 = 0
2x = 1 ou 2x = -1
x = 1/2 ou x = -1/2
25x^2 - 100 = 0
(5x)^2 - 10^2 = 0
(5x - 10)(5x + 10) = 0
5(x - 2) * 5(x + 2) = 0
25(x - 2)(x + 2) = 0
x - 2 = 0 ou x + 2 = 0
x = 2 ou x = -2
5x^2 - 3x = 0
5x * x - 3 * x = 0
x(5x - 3) = 0
x = 0 ou 5x - 3 = 0
x = 0 ou 5x = 3
x = 0 ou x = 3/5
Exercice 74 :
(2x + 3)(x - 2) + 2(x - 2) = 0
(x - 2)(2x + 3 + 2) = 0
(x - 2)(2x + 5) = 0
x - 2 = 0 ou 2x + 5 = 0
x = 2 ou 2x = -5
x = 2 ou x = -5/2
25x^2 - 30x + 9 = 0
(5x)^2 - 2 * 5x * 3 + 3^2 = 0
(5x - 3)^2 = 0
5x - 3 = 0
5x = 3
x = 3/5
(3x + 1)(5x - 1) - 4(5x - 1) = 0
(5x - 1)(3x + 1 - 4) = 0
(5x - 1)(3x - 3) = 0
(5x - 1) * 3(x - 1) = 0
3(5x - 1)(x - 1) = 0
5x - 1 = 0 ou x - 1 = 0
5x = 1 ou x = 1
x = 1/5 ou x = 1
(3x + 1)^2 - 49 = 0
(3x + 1)^2 - 7^2 = 0
(3x + 1 - 7)(3x + 1 + 7) = 0
(3x - 6)(3x + 8) = 0
3(x - 2)(3x + 8) = 0
x - 2 = 0 ou 3x + 8 = 0
x = 2 ou 3x = -8
x = 2 ou x = -8/3
Exercice 75
x(2x - 5) = 2(2x - 5)
x(2x - 5) - 2(2x - 5) = 0
(2x - 5)(x - 2) = 0
2x - 5 = 0 ou x - 2 = 0
2x = 5 ou x = 2
x = 5/2 ou x = 2
(5x + 2)(2x - 3) = 4x^2 - 12x + 9
(5x + 2)(2x - 3) = (2x)^2 - 2 * 2x * 3 + 3^2
(5x + 2)(2x - 3) = (2x - 3)^2
(5x + 2)(2x - 3) - (2x - 3)^2 = 0
(2x - 3)(5x + 2 - 2x + 3) = 0
(2x - 3)(3x + 5) = 0
2x - 3 = 0 ou 3x + 5 = 0
2x = 3 ou 3x = -5
x = 3/2 ou x = -5/3
5x^2 = 8x(x + 1)
8x^2 + 8x - 5x^2 = 0
3x^2 + 8x = 0
x * 3x + x * 8 = 0
x(3x + 8) = 0
x = 0 ou 3x + 8 = 0
x = 0 ou 3x = -8
x = 0 ou x = -8/3
(2x + 1)^2 = 9
(2x + 1)^2 - 3^2 = 0
(2x + 1 - 3)(2x + 1 + 3) = 0
(2x - 2)(2x + 4) = 0
2(x - 1) * 2(x + 2) = 0
4(x - 1)(x + 2) = 0
x - 1 = 0 ou x + 2 = 0
x = 1 ou x = -2
(4x + 7)^2 = (x - 2)^2
(4x + 7)^2 - (x - 2)^2 = 0
(4x + 7 - x + 2)(4x + 7 + x - 2) = 0
(3x + 9)(5x + 5) = 0
3(x + 3) * 5(x + 1) = 0
15(x + 3)(x + 1) = 0
x + 3 = 0 ou x + 1 = 0
x = -3 ou x = -1
(7x - 2)^2 - 1 = 80
(7x - 2)^2 - 1 - 80 = 0
(7x - 2)^2 - 81 = 0
(7x - 2)^2 - 9^2 = 0
(7x - 2 - 9)(7x - 2 + 9) = 0
(7x - 11)(7x + 7) = 0
(7x - 11) * 7(x + 1) = 0
7(7x - 11)(x + 1) = 0
7x - 11 = 0 ou x + 1 = 0
7x = 11 ou x = -1
x = 11/7 ou x = -1
