Mathématiques — Première

Polynômes du second degré — Première Maths

Discriminant, racines, factorisation, forme canonique et équations du second degré.

Par ProfBot

Polynômes du second degré — Première Mathématiques

🎯 Objectif : Résoudre toute équation et inéquation du second degré.

1. Forme générale


f(x) = ax² + bx + c    (a ≠ 0)

Représentation : une parabole

a > 0 → parabole ouverte vers le haut (∪)
a < 0 → parabole ouverte vers le bas (∩)

2. Le discriminant Δ


Δ = b² − 4ac

Δ	Racines	Factorisation
Δ > 0	x₁ = (−b−√Δ)/(2a) et x₂ = (−b+√Δ)/(2a)	a(x−x₁)(x−x₂)
Δ = 0	x₀ = −b/(2a) (double)	a(x−x₀)²
Δ < 0	Aucune racine réelle	Irréductible

3. Exemple graphique

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</svg>

f(x) = x²−2x−3, a=1, b=−2, c=−3 :


Δ = (−2)² − 4×1×(−3) = 4+12 = 16 > 0

x₁ = (2−4)/2 = −1    x₂ = (2+4)/2 = 3

Sommet : α = −b/(2a) = 1,  β = f(1) = 1−2−3 = −4
→ S(1 ; −4)

4. Forme canonique


f(x) = a(x − α)² + β

α = −b/(2a)    β = f(α) = −Δ/(4a)

5. Signe du trinôme

Si Δ > 0 (racines x₁ < x₂) :

x	]−∞ ; x₁[	x₁	]x₁ ; x₂[	x₂	]x₂ ; +∞[
f(x)	signe a	0	signe −a	0	signe a

💡 f(x) est du signe de a à l'extérieur des racines, du signe de −a entre elles.

6. Relations de Viète


x₁ + x₂ = −b/a
x₁ × x₂ = c/a

📝 Exercices

Résoudre 2x²−5x+3=0 et factoriser.
Signe de f(x)=−x²+4x−4.
Forme canonique de f(x)=x²−6x+7. Coordonnées du sommet ?
Résoudre 3x²+x−2≥0.

✅ À retenir


Δ = b²−4ac
Δ>0 → 2 racines : (−b±√Δ)/(2a)
Δ=0 → 1 racine double : x=−b/(2a)
Δ<0 → pas de racine réelle
Signe : même signe que a hors des racines

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