Solution Série d'exercices : Équation et inéquation du 2nd degré - 2nd L

 Exercice 1 : Forme Canonique

La forme canonique d'une fonction polynôme du second degré \( f(x) = ax^2 + bx + c \) est donnée par :
\[ f(x) = a\left( (x - \alpha)^2 \right) + \beta \]
où \( \alpha = -\dfrac{b}{2a} \) et \( \beta = f(\alpha) \).

 1. \( f(x) = x^2 - 4x + 3 \)
\( a = 1 \), \( b = -4 \), \( c = 3 \)
 \( \alpha = -\dfrac{-4}{2 \times 1} = 2 \)
 \( \beta = f(2) = (2)^2 - 4 \times 2 + 3 = 4 - 8 + 3 = -1 \)
 Forme canonique : \( f(x) = (x - 2)^2 - 1 \)

 2. \( f(x) = 2x^2 - 3x + 7 \)
 \( a = 2 \), \( b = -3 \), \( c = 7 \)
 \( \alpha = -\dfrac{-3}{2 \times 2} = \dfrac{3}{4} \)
 \( \beta = f\left(\dfrac{3}{4}\right) = 2 \left(\dfrac{3}{4}\right)^2 - 3 \times \dfrac{3}{4} + 7 = 2 \times \dfrac{9}{16} - \dfrac{9}{4} + 7 = \dfrac{9}{8} - \dfrac{18}{8} + \dfrac{56}{8} = \dfrac{47}{8} \)
 Forme canonique : \( f(x) = 2\left(x - \dfrac{3}{4}\right)^2 + \dfrac{47}{8} \)

 3. \( f(x) = \dfrac{1}{2}x^2 - 5x - 1 \)
 \( a = \dfrac{1}{2} \), \( b = -5 \), \( c = -1 \)
 \( \alpha = -\dfrac{-5}{2 \times \dfrac{1}{2}} = 5 \)
 \( \beta = f(5) = \dfrac{1}{2} \times 25 - 5 \times 5 - 1 = \dfrac{25}{2} - 25 - 1 = \dfrac{25}{2} - \dfrac{50}{2} - \dfrac{2}{2} = -\dfrac{27}{2} \)
 Forme canonique : \( f(x) = \dfrac{1}{2}(x - 5)^2 - \dfrac{27}{2} \)

 4. \( f(x) = 169x^2 + 13x - 1 \)
 \( a = 169 \), \( b = 13 \), \( c = -1 \)
 \( \alpha = -\dfrac{13}{2 \times 169} = -\dfrac{13}{338} = -\dfrac{1}{26} \)
 \( \beta = f\left(-\dfrac{1}{26}\right) = 169 \left(-\dfrac{1}{26}\right)^2 + 13 \times \left(-\dfrac{1}{26}\right) - 1 = 169 \times \dfrac{1}{676} - \dfrac{13}{26} - 1 = \dfrac{1}{4} - \dfrac{1}{2} - 1 = -\dfrac{5}{4} \)
 Forme canonique : \( f(x) = 169\left(x + \dfrac{1}{26}\right)^2 - \dfrac{5}{4} \)

 5. \( f(x) = \sqrt{3}x^2 - (1 - \sqrt{3})x + 4 \)
 \( a = \sqrt{3} \), \( b = -(1 - \sqrt{3}) \), \( c = 4 \)
 \( \alpha = -\dfrac{-(1 - \sqrt{3})}{2 \times \sqrt{3}} = \dfrac{1 - \sqrt{3}}{2\sqrt{3}} = \dfrac{\sqrt{3} - 3}{6} \) (en rationalisant)
 \( \beta = f(\alpha) \) : Calcul complexe, la forme canonique est :
  \[ f(x) = \sqrt{3}\left(x - \dfrac{1 - \sqrt{3}}{2\sqrt{3}}\right)^2 + \beta \]
 

 6. \( f(x) = x^2 + 2x + 2 \)
- \( a = 1 \), \( b = 2 \), \( c = 2 \)
- \( \alpha = -\dfrac{2}{2 \times 1} = -1 \)
- \( \beta = f(-1) = (-1)^2 + 2 \times (-1) + 2 = 1 - 2 + 2 = 1 \)
- Forme canonique : \( f(x) = (x + 1)^2 + 1 \)

 7. \( f(x) = 3x^2 + 12x - 7 \)
 \( a = 3 \), \( b = 12 \), \( c = -7 \)
 \( \alpha = -\dfrac{12}{2 \times 3} = -2 \)
 \( \beta = f(-2) = 3 \times 4 + 12 \times (-2) - 7 = 12 - 24 - 7 = -19 \)
 Forme canonique : \( f(x) = 3(x + 2)^2 - 19 \)

 8. \( f(x) = -x^2 + x + 2 \)
 \( a = -1 \), \( b = 1 \), \( c = 2 \)
 \( \alpha = -\dfrac{1}{2 \times (-1)} = \dfrac{1}{2} \)
 \( \beta = f\left(\dfrac{1}{2}\right) = -\left(\dfrac{1}{2}\right)^2 + \dfrac{1}{2} + 2 = -\dfrac{1}{4} + \dfrac{1}{2} + 2 = \dfrac{9}{4} \)
 Forme canonique : \( f(x) = -\left(x - \dfrac{1}{2}\right)^2 + \dfrac{9}{4} \)

 9. \( f(x) = -3x^2 + 7x - 2 \)
 \( a = -3 \), \( b = 7 \), \( c = -2 \)
 \( \alpha = -\dfrac{7}{2 \times (-3)} = \dfrac{7}{6} \)
 \( \beta = f\left(\dfrac{7}{6}\right) = -3 \left(\dfrac{7}{6}\right)^2 + 7 \times \dfrac{7}{6} - 2 = -3 \times \dfrac{49}{36} + \dfrac{49}{6} - 2 = -\dfrac{49}{12} + \dfrac{98}{12} - \dfrac{24}{12} = \dfrac{25}{12} \)
 Forme canonique : \( f(x) = -3\left(x - \dfrac{7}{6}\right)^2 + \dfrac{25}{12} \)

 Exercice 2 : Calcul du Discriminant

Le discriminant \( \Delta \) d'une fonction \( f(x) = ax^2 + bx + c \) est donné par :
\[ \Delta = b^2 - 4ac \]

 1. \( f(x) = x^2 - 4x + 3 \)
 \( \Delta = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4 \)

 2. \( f(x) = 2x^2 - 3x + 7 \)
 \( \Delta = (-3)^2 - 4 \times 2 \times 7 = 9 - 56 = -47 \)

 3. \( f(x) = \dfrac{1}{2}x^2 - 5x - 1 \)
 \( \Delta = (-5)^2 - 4 \times \dfrac{1}{2} \times (-1) = 25 + 2 = 27 \)

 4. \( f(x) = 169x^2 + 13x - 1 \)
- \( \Delta = 13^2 - 4 \times 169 \times (-1) = 169 + 676 = 845 \)

 5. \( f(x) = \sqrt{3}x^2 - (1 - \sqrt{3})x + 4 \)
 \( \Delta = (1 - \sqrt{3})^2 - 4 \times \sqrt{3} \times 4 = 1 - 2\sqrt{3} + 3 - 16\sqrt{3} = 4 - 18\sqrt{3} \)

 6. \( f(x) = x^2 + 2x + 2 \)
 \( \Delta = 2^2 - 4 \times 1 \times 2 = 4 - 8 = -4 \)

 7. \( f(x) = 3x^2 + 12x - 7 \)
 \( \Delta = 12^2 - 4 \times 3 \times (-7) = 144 + 84 = 228 \)

 8. \( f(x) = -x^2 + x + 2 \)
 \( \Delta = 1^2 - 4 \times (-1) \times 2 = 1 + 8 = 9 \)

 9. \( f(x) = -3x^2 + 7x - 2 \)
 \( \Delta = 7^2 - 4 \times (-3) \times (-2) = 49 - 24 = 25 \)

Exercice 3 : Résolution d'Équations

Pour résoudre \( ax^2 + bx + c = 0 \), on utilise :
\[ x = \dfrac{-b \pm \sqrt{\Delta}}{2a} \]

 1. \( x^2 + 3x - 2 = 0 \)
 \( \Delta = 9 + 8 = 17 \)
 Solutions : \( x = \dfrac{-3 \pm \sqrt{17}}{2} \)

 2. \( x^2 + 4x - 21 = 0 \)
 \( \Delta = 16 + 84 = 100 \)
 Solutions : \( x = \dfrac{-4 \pm 10}{2} \) soit \( x = 3 \) ou \( x = -7 \)

 3. \( 6x^2 - x - 5 = 0 \)
 \( \Delta = 1 + 120 = 121 \)
 Solutions : \( x = \dfrac{1 \pm 11}{12} \) soit \( x = 1 \) ou \( x = -\dfrac{5}{6} \)

 4. \( x^2 + x + 1 = 0 \)
 \( \Delta = 1 - 4 = -3 \)
 Aucune solution réelle.

 5. \( 2x^2 + 2x - 7 = 0 \)
 \( \Delta = 4 + 56 = 60 \)
 Solutions : \( x = \dfrac{-2 \pm 2\sqrt{15}}{4} = \dfrac{-1 \pm \sqrt{15}}{2} \)

 8. \( \sqrt{2}x^{2} - (1 - \sqrt{2})x - 1 = 0 \)
 \( \Delta = (1 - \sqrt{2})^2 + 4\sqrt{2} = 1 - 2\sqrt{2} + 2 + 4\sqrt{2} = 3 + 2\sqrt{2} \)
 Solutions : \( x = \dfrac{1 - \sqrt{2} \pm \sqrt{3 + 2\sqrt{2}}}{2\sqrt{2}} \)

 9. \( 3x^{2} + 12x - 7 = 0 \)
 \( \Delta = 144 + 84 = 228 \)
 Solutions : \( x = \dfrac{-12 \pm \sqrt{228}}{6} = \dfrac{-12 \pm 2\sqrt{57}}{6} = \dfrac{-6 \pm \sqrt{57}}{3} \)

 10. \( -x^{2} + x + 2 = 0 \)
- \( \Delta = 1 + 8 = 9 \)
- Solutions : \( x = \dfrac{-1 \pm 3}{-2} \) soit \( x = 2 \) ou \( x = -1 \)

 11. \( -2x^{2} - 2x + 5 = 0 \)
 \( \Delta = 4 + 40 = 44 \)
 Solutions : \( x = \dfrac{2 \pm \sqrt{44}}{-4} = \dfrac{-2 \mp \sqrt{44}}{4} = \dfrac{-1 \mp \sqrt{11}}{2} \)

Exercice 4 : Somme et Produit

 1. Déterminer deux nombres réels de somme \( S \) et produit \( P \)
 Ils existent si \( \Delta = S^2 - 4P \geq 0 \).

a. \( S = 3 \), \( P = -10 \)
 \( \Delta = 9 + 40 = 49 \geq 0 \)
 Solutions : \( x^2 - 3x - 10 = 0 \) soit \( x = \dfrac{3 \pm 7}{2} \) donc \( 5 \) et \( -2 \).

b. \( S = 5 \), \( P = 6 \)
 \( \Delta = 25 - 24 = 1 \geq 0 \)
 Solutions : \( x^2 - 5x + 6 = 0 \) soit \( x = \dfrac{5 \pm 1}{2} \) donc \( 3 \) et \( 2 \).

c. \( S = -6 \), \( P = 9 \)
 \( \Delta = 36 - 36 = 0 \)
 Solution double : \( x^2 + 6x + 9 = 0 \) soit \( x = -3 \).

 2. Somme 9 et produit -70
 \( \Delta = 81 + 280 = 361 \geq 0 \)
 Solutions : \( x^2 - 9x - 70 = 0 \) soit \( x = \dfrac{9 \pm 19}{2} \) donc \( 14 \) et \( -5 \).

 3. Pour \( x^2 - 3x + 2 = 0 \)
a. \( x_1 + x_2 = 3 \)

b. \( x_1 \times x_2 = 2 \)

c. \( \dfrac{1}{x_1} + \dfrac{1}{x_2} = \dfrac{x_1 + x_2}{x_1 x_2} = \dfrac{3}{2} \)

d. \( x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = 9 - 4 = 5 \)

Exercice 5 : Somme et Produit (Suite)

 1. Équations à résoudre
a. \( x^2 - 2x + 7 = 0 \)
 \( \Delta = 4 - 28 = -24 \)
 Aucune solution réelle.

b. \( 6x^2 - x - 5 = 0 \)
- Solutions : \( x = 1 \) et \( x = -\dfrac{5}{6} \).

c. \( 8x^2 + x + 1 = 0 \)
 \( \Delta = 1 - 32 = -31 \)
 Aucune solution réelle.

d. \( \sqrt{2}x^2 - (1 + \sqrt{2})x - 1 = 0 \)
 \( \Delta = (1 + \sqrt{2})^2 + 4\sqrt{2} = 1 + 2\sqrt{2} + 2 + 4\sqrt{2} = 3 + 6\sqrt{2} \)
 Solutions : \( x = \dfrac{1 + \sqrt{2} \pm \sqrt{3 + 6\sqrt{2}}}{2\sqrt{2}} \)

 2. Rectangle de surface 861 et périmètre 124
 Soit \( x \) et \( y \) les dimensions.
 \( x + y = 62 \) et \( xy = 861 \).
 Solutions : \( x^2 - 62x + 861 = 0 \)
 \( \Delta = 3844 - 3444 = 400 \)
 \( x = \dfrac{62 \pm 20}{2} \) soit \( 41 \) et \( 21 \).

 3. Résolution mentale
a. \( 3x^2 + 7x - 10 = 0 \)
 Somme des coefficients : \( 3 + 7 - 10 = 0 \) donc \( x = 1 \) est racine.
 L'autre racine est \( -\dfrac{10}{3} \).

b. \( 2x^2 + 9x + 7 = 0 \)
 Somme des coefficients : \( 2 + 9 + 7 = 18 \neq 0 \).
 Produit des racines : \( \dfrac{7}{2} \), somme \( -\dfrac{9}{2} \).
 Racines : \( -1 \) et \( -\dfrac{7}{2} \).

 4. Vérification de \( 2 \) comme racine de \( x^4 + 11x - 26 = 0 \)
 \( 2^4 + 11 \times 2 - 26 = 16 + 22 - 26 = 12 \neq 0 \).
 Erreur dans l'énoncé.

Exercice 6 : Systèmes d'Équations

 \( S_1 \) : \( x + y = 13 \), \( xy = 40 \)
 Solutions : \( t^2 - 13t + 40 = 0 \) soit \( t = 8 \) et \( t = 5 \).
 \( (8, 5) \) et \( (5, 8) \).

 \( S_2 \) : \( x + y = -1 \), \( xy = -1 \)
 Solutions : \( t^2 + t - 1 = 0 \) soit \( t = \dfrac{-1 \pm \sqrt{5}}{2} \).
 \( \left( \dfrac{-1 + \sqrt{5}}{2}, \dfrac{-1 - \sqrt{5}}{2} \right) \) et réciproque.

 \( S_3 \) : \( x + y = 2 \), \( xy = 3 \)
 \( \Delta = 4 - 12 = -8 \).
 Aucune solution réelle.

 \( S_4 \) : \( x + y = 4 \), \( xy = -12 \)
 Solutions : \( t^2 - 4t - 12 = 0 \) soit \( t = 6 \) et \( t = -2 \).
 \( (6, -2) \) et \( (-2, 6) \).

 \( S_5 \) : \( x + y = 5 \), \( x^2 + y^2 = 13 \)
 \( x^2 + y^2 = (x + y)^2 - 2xy \) donc \( 13 = 25 - 2xy \) soit \( xy = 6 \).
- Solutions : \( t^2 - 5t + 6 = 0 \) soit \( t = 2 \) et \( t = 3 \).
- \( (2, 3) \) et \( (3, 2) \).

 \( S_6 \) : \( xy = -2 \), \( x^2 + y^2 = 5 \)
 \( x^2 + y^2 = (x + y)^2 - 2xy \) donc \( 5 = (x + y)^2 + 4 \) soit \( (x + y)^2 = 1 \).
 Deux cas :
  1. \( x + y = 1 \), \( xy = -2 \) : \( t^2 - t - 2 = 0 \) soit \( t = 2 \) et \( t = -1 \).
  2. \( x + y = -1 \), \( xy = -2 \) : \( t^2 + t - 2 = 0 \) soit \( t = 1 \) et \( t = -2 \).
 Solutions : \( (2, -1) \), \( (-1, 2) \), \( (1, -2) \), \( (-2, 1) \).

 \( S_7 \) : \( x + y = 9 \), \( x^2 + y^2 = 25 \)
 \( 25 = 81 - 2xy \) donc \( xy = 28 \).
 Solutions : \( t^2 - 9t + 28 = 0 \), \( \Delta = 81 - 112 = -31 \).
 Aucune solution réelle.

 \( S_8 \) : \( x + y = 4 \), \( x^2 + y^2 = 10 \)
 \( 10 = 16 - 2xy \) donc \( xy = 3 \).
 Solutions : \( t^2 - 4t + 3 = 0 \) soit \( t = 1 \) et \( t = 3 \).
 \( (1, 3) \) et \( (3, 1) \).

Exercice 7 : Factorisation

 1. \( f(x) = 4x^2 - 4x + 1 \)
 \( \Delta = 16 - 16 = 0 \), racine double \( x = \dfrac{4}{8} = \dfrac{1}{2} \).
 \( f(x) = 4\left(x - \dfrac{1}{2}\right)^2 \).

 2. \( f(x) = -x^2 + 4x + 30 \)
 \( \Delta = 16 + 120 = 136 \), racines \( x = \dfrac{-4 \pm \sqrt{136}}{-2} = 2 \mp \sqrt{34} \).
 \( f(x) = -\left(x - 2 + \sqrt{34}\right)\left(x - 2 - \sqrt{34}\right) \).

 3. \( f(x) = x^2 - 5x - 1 \)
 \( \Delta = 25 + 4 = 29 \), racines \( x = \dfrac{5 \pm \sqrt{29}}{2} \).
 \( f(x) = \left(x - \dfrac{5 + \sqrt{29}}{2}\right)\left(x - \dfrac{5 - \sqrt{29}}{2}\right) \).

 4. \( f(x) = 5x^2 - 15x - 20 \)
 Simplifier par 5 : \( x^2 - 3x - 4 \), \( \Delta = 9 + 16 = 25 \), racines \( x = 4 \) et \( x = -1 \).
 \( f(x) = 5(x - 4)(x + 1) \).

 5. \( f(x) = x^2 - \sqrt{3}x + 4 \)
 \( \Delta = 3 - 16 = -13 \).
 Pas de factorisation réelle.

 6. \( f(x) = -3x^2 + 4x + 3 \)
 \( \Delta = 16 + 36 = 52 \), racines \( x = \dfrac{-4 \pm \sqrt{52}}{-6} = \dfrac{2 \mp \sqrt{13}}{3} \).
 \( f(x) = -3\left(x - \dfrac{2 + \sqrt{13}}{3}\right)\left(x - \dfrac{2 - \sqrt{13}}{3}\right) \).

 7. \( f(x) = -9x^2 + 12x - 4 \)
 \( \Delta = 144 - 144 = 0 \), racine double \( x = \dfrac{-12}{-18} = \dfrac{2}{3} \).
 \( f(x) = -9\left(x - \dfrac{2}{3}\right)^2 \).

 8. \( f(x) = -15x^2 + 11x - 2 \)
 \( \Delta = 121 - 120 = 1 \), racines \( x = \dfrac{-11 \pm 1}{-30} \) soit \( \dfrac{2}{5} \) et \( \dfrac{1}{3} \).
 \( f(x) = -15\left(x - \dfrac{2}{5}\right)\left(x - \dfrac{1}{3}\right) \).

 9. \( f(x) = 2x^2 - x + 1 \)
 \( \Delta = 1 - 8 = -7 \).
 Pas de factorisation réelle.

Exercice 8 : Factorisation

 1. \( f(x) = x^2 + 4x + 4 \)
- Carré parfait : \( f(x) = (x + 2)^2 \).

 2. \( g(x) = (x + 2)(2x - 4) + x^2 + 4x + 4 \)
 Développer : \( 2x^2 - 4x + 4x - 8 + x^2 + 4x + 4 = 3x^2 + 4x - 4 \).
 \( \Delta = 16 + 48 = 64 \), racines \( x = \dfrac{-4 \pm 8}{6} \) soit \( \dfrac{2}{3} \) et \( -2 \).
 \( g(x) = 3\left(x - \dfrac{2}{3}\right)(x + 2) \).

Exercice 9 : Inéquations

 1. \( x^2 - 2x - 1 < 0 \)
 Racines : \( x = 1 \pm \sqrt{2} \).
 Solution : \( x \in ]1 - \sqrt{2}, 1 + \sqrt{2}[ \).

 2. \( 3x^2 - 5x + 22 \leq 0 \)
 \( \Delta = 25 - 264 = -239 \), toujours positif (car \( a = 3 > 0 \)).
 Solution : \( \emptyset \).

 3. \( 9x^2 + 6x + 1 \leq 0 \)
 Carré parfait : \( (3x + 1)^2 \leq 0 \).
 Solution : \( x = -\dfrac{1}{3} \).

 4. \( 4x^2 - 4x - 1 \geq 0 \)
Racines : \( x = \dfrac{4 \pm \sqrt{32}}{8} = \dfrac{1 \pm \sqrt{2}}{2} \).
Solution : \( x \leq \dfrac{1 - \sqrt{2}}{2} \) ou \( x \geq \dfrac{1 + \sqrt{2}}{2} \).

 5. \( 2x^2 + 2x - 7 < 0 \)
 Racines : \( x = \dfrac{-2 \pm \sqrt{60}}{4} = \dfrac{-1 \pm \sqrt{15}}{2} \).
 Solution : \( x \in \left]\dfrac{-1 - \sqrt{15}}{2}, \dfrac{-1 + \sqrt{15}}{2}\right[ \).

 6. \( 6x^2 - x - 5 < 0 \)
 Racines : \( x = 1 \) et \( x = -\dfrac{5}{6} \).
 Solution : \( x \in \left]-\dfrac{5}{6}, 1\right[ \).

 7. \( x^2 + x + 1 \geq 0 \)
 \( \Delta = 1 - 4 = -3 \), toujours positif.
 Solution : \( \mathbb{R} \).

 8. \( x^2 - x - 1 > 0 \)
 Racines : \( x = \dfrac{1 \pm \sqrt{5}}{2} \).
 Solution : \( x < \dfrac{1 - \sqrt{5}}{2} \) ou \( x > \dfrac{1 + \sqrt{5}}{2} \).

 9. \( (x^2 + x + 1)^2 < (x^2 - x + 1)^2 \)
 Différence de carrés : \( (2x)(2x^2 + 2) < 0 \) soit \( 4x(x^2 + 1) < 0 \).
 Solution : \( x < 0 \).

 10. \( (-3x^2 + 4x - 2)^2 \geq (3x^2 - x + 5)^2 \)
 Différence de carrés : \( (-6x^2 + 5x - 7)(3x - 7) \geq 0 \).
 Solution : \( x \leq \dfrac{7}{3} \) (car \( -6x^2 + 5x - 7 \) toujours négatif).

Exercice 10 : Problème de Bateau

 1. Vitesse par rapport à la rive
 Aller (descendre) : \( v + 5 \) km/h.
 Retour (remonter) : \( v - 5 \) km/h.

 2. Durée des trajets
 Aller : \( t_1 = \dfrac{75}{v + 5} \) heures.
 Retour : \( t_2 = \dfrac{75}{v - 5} \) heures.
 Total : \( t_1 + t_2 = 8 \) heures.

 3. Calcul de \( v \)
\[ \dfrac{75}{v + 5} + \dfrac{75}{v - 5} = 8 \]
\[ 75(v - 5) + 75(v + 5) = 8(v^2 - 25) \]
\[ 150v = 8v^2 - 200 \]
\[ 8v^2 - 150v - 200 = 0 \]
Simplifier par 2 :
\[ 4v^2 - 75v - 100 = 0 \]
\[ \Delta = 5625 + 1600 = 7225 \]
\[ v = \dfrac{75 \pm 85}{8} \]
Solution positive : \( v = \dfrac{160}{8} = 20 \) km/h.

Réponse : La vitesse propre du bateau est \( 20 \) km/h.