Exercice : soit f(x) = 2x² - 4x - 6 1) Verifier que f(x) = 2 (x-1)² - 8 2) Montrer que f(x) = 2 ( x-3) (x+1) 3) Choisir la bonne ecriture pour calculer f(3) ; f

Bonsoir

1) Verifions que f(x) = 2 (x-1)² - 8

Développons f(x)
2 (x-1)² - 8 = 2(x²-2x+1) - 8 = 2x² - 4x + 2 - 8 = 2x² - 4x - 6
nous avons la même égalité

2) Montrer que f(x) = 2 ( x-3) (x+1)

2x² - 4x - 6 = 2(x²-2x-3)

Factorisons
x²-2x-3
Calculons Δ
Δ = (-2)² - 4(1)(-3) = 4 + 12 = 16

Posons
2(x²-2x-3) = 0
Déterminons les racines

x1 = (2 + 4) / 2 = 3
x2 = (2 - 4) / 2 = - 1

donc f(x) = 2(x-3)(x+1)

3) Choisir la bonne ecriture pour calculer
2x² - 4x - 6
f(3) = 2(3)² - 4(3) - 6 = 18 - 12 - 6 = 0
f(1) = 2(1)² - 4(1) - 6 = 2 - 4 - 6 = -8
f(√2) = 2(√2)² - 4(√2) - 6 = 4 - 4√2 - 6 = - 2 - 4√2

4) Résoudre
f(x) = 0

2x² - 4x - 6 = 0
Δ = (-4)² - 4(2)(-6) = 16 + 48 = 64
√Δ = 8

x1 = (4 - 8) / 4 = -1
x2 = (4+8) / 4 = 3
S = {-1;3}

f(x) = - 8
2x² - 4x - 6 = - 8
2x² - 4x - 6 + 8 = 0
2x² - 4x + 2 = 0
on simplifie par 2
x² - 2x + 1 = (x+1)²
x+1 = 0
     x = -1
S = {-1}


f(x) = - 6

2x² - 4x - 6 = - 6
2x² - 4x - 6 + 6 = 0
2x² - 4x = 0
2x(x-2) = 0

2x = 0        ou            x - 2 = 0
x = 0                              x = 2

S ={0;2}