Développement :

h(x) = x² – 9 – (3 – x)(x + 5)

= x² – 9 – [3x + 15 – x² – 5x]

= x² – 9 – (-x² – 2x + 15)

= x² – 9 + x² + 2x – 15

= 2x² + 2x – 24

h(x) = 2x² + 2x – 24 = 2(x² + x – 12)

Factorisation du trinôme : x² + x – 12 = (x + 4)(x – 3)

h(x) = 2(x + 4)(x – 3)

• h(-3) = 2(-3 + 4)(-3 – 3) = 2(1)(-6) = -12

• h(3/4) = 2(3/4 + 4)(3/4 – 3) = 2(19/4)(-9/4) = -171/8

• h(1 – √2) = 2(5 – √2)(-2 – √2) = 2[-8 – 3√2] = -16 – 6√2

• h(x) = 0 ⇒ 2(x + 4)(x – 3) = 0 ⇒ x = -4 ou x = 3

• h(x) = -24 ⇒ 2x² + 2x – 24 = -24 ⇒ 2x(x + 1) = 0 ⇒ x = 0 ou x = -1

• h(x) ≥ 0 : Tableau de signes

Solution : x ∈ ]-∞, -4] ∪ [3, +∞[

• Domaine : h(x) ≠ 0 ⇒ x ≠ -4 et x ≠ 3

• (f×g)(x) = (3x – 1)(-x + 3) = -3x² + 10x – 3

r(x) = (-3x² + 10x – 3)/[2(x + 4)(x – 3)]

2√3 ≈ 3.464

r(3.464) ≈ [-3(12) + 10(3.464) – 3]/[2(7.464)(0.464)] ≈ -4.36/6.92 ≈ -0.63

-0.64 < r(2√3) < -0.62

⎧ 3x + 2y + 2 = 0

⎨ y/x + 4 = 0 ⇒ y = -4x

Substitution : 3x + 2(-4x) + 2 = 0 ⇒ -5x + 2 = 0 ⇒ x = 2/5

Solution : (2/5, -8/5)

• Tracer les droites :

– 3x + 2y + 2 = 0

– y = -4x

Point d’intersection : (2/5, -8/5)

a) √x² = |x|

b) Si |x| = |y| alors x = y ou x = -y

√2 ≈ 1.414 ⇒ 3√2 ≈ 4.242 ⇒ M ≈ -0.242 < 0

M² = (4 – 3√2)² = 16 – 24√2 + 18 = 34 – 24√2

N² = (2 + (3/2)√2)² = 4 + 6√2 + 9/2 = 8.5 + 6√2

√(34 – 24√2) = a√2 + b ⇒ 34 – 24√2 = 2a² + b² + 2ab√2

Par identification : a = 3, b = -4

Z = 3√2 – 4

M² + 4N² = (34 – 24√2) + 4(8.5 + 6√2) = 34 + 34 = 68

A(x) = 16 – (4x² – 28x + 49) = -4x² + 28x – 33

A(x) = 16 – (2x – 7)² = (4 – (2x – 7))(4 + (2x – 7)) = (-2x + 11)(2x – 3)

B(x) = (x – 2)² – (x – 1)(2 – x) = (x – 2)(2x – 3)

a) (-2x + 11)(2x – 3) = 0 ⇒ x = 11/2 ou x = 3/2

b) (x – 2)(2x – 3) ≥ 0

Solution : x ∈ ]-∞, 3/2] ∪ [2, +∞[

-4x² + 28x – 33 < x² – 4x + 4 – (2x – x² – 2 + x)

Simplification et résolution de l’inéquation…