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Populaire Trigonométrie >

arcsin(6x)+arcsin(6sqrt(3x))=-pi/2

Solution

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

Solution

Aucune solution pour x \in R

étapes des solutions

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

Récrire en utilisant des identités trigonométriques

arcsin (6 x) + arcsin (6 3 x)

Utiliser l'identité de la somme au produit:

arcsin (s) + arcsin (t) = arcsin (s 1 - t^{2} + t 1 - s^{2})

= arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2})

arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2}) = - \frac{π}{2}

Appliquer les propriétés trigonométriques inverses

arcsin (6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2}) = - \frac{π}{2}

arcsin (x) = a \Rightarrow x = sin (a)

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = sin (- \frac{π}{2})

sin (- \frac{π}{2}) = - 1

sin (- \frac{π}{2})

Utiliser la propriété suivante :

sin (- x) = - sin (x)

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= - sin (\frac{π}{2})

Utiliser l'identité triviale suivante:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

Tableau de périodicité

sin (x)

avec un cycle

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= - 1

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

Résoudre

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1 :

Aucune solution pour

x \in R

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

Supprimer les racines carrées

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

Soustraire

6 3 x 1 - (6 x)^{2}

des deux côtés

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} - 6 3 x 1 - (6 x)^{2} = - 1 - 6 3 x 1 - (6 x)^{2}

Simplifier

6 1 - (6 3 x)^{2} x = - 1 - 6 3 x 1 - (6 x)^{2}

Mettre les deux côtés au carré:

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

(6 1 - (6 3 x)^{2} x)^{2} = (- 1 - 6 3 x 1 - (6 x)^{2})^{2}

Développer

(6 1 - (6 3 x)^{2} x)^{2} : 36 x^{2} - 3888 x^{3}

(6 1 - (6 3 x)^{2} x)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} x^{2} (1 - (6 3 x)^{2})^{2}

(1 - (6 3 x)^{2})^{2} : 1 - (6 3 x)^{2}

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((1 - (6 3 x)^{2})^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= (1 - (6 3 x)^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 1 - (6 3 x)^{2}

= 6^{2} (1 - (6 3 x)^{2}) x^{2}

6^{2} = 36

= 36 (1 - (6 3 x)^{2}) x^{2}

Développer

36 (1 - (6 3 x)^{2}) x^{2} : 36 x^{2} - 3888 x^{3}

36 (1 - (6 3 x)^{2}) x^{2}

(6 3 x)^{2} = 6^{2} \cdot 3 x

(6 3 x)^{2}

3 x = 3 x

3 x

Appliquer la règle des radicaux :

n ab = n a n b,

en supposant

a \geq 0, b \geq 0

= 3 x

= (63 x)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} (3)^{2} (x)^{2}

(3)^{2} : 3

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 3

= 6^{2} \cdot 3 (x)^{2}

(x)^{2} : x

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (x^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= x^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= x

= 6^{2} \cdot 3 x

= 36 x^{2} (- 6^{2} \cdot 3 x + 1)

6^{2} \cdot 3 x = 108 x

6^{2} \cdot 3 x

6^{2} = 36

= 36 \cdot 3 x

Multiplier les nombres :

36 \cdot 3 = 108

= 108 x

= 36 x^{2} (- 108 x + 1)

= 36 x^{2} (1 - 108 x)

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = 36 x^{2}, b = 1, c = 108 x

= 36 x^{2} \cdot 1 - 36 x^{2} \cdot 108 x

= 36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x

Simplifier

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x : 36 x^{2} - 3888 x^{3}

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x

36 \cdot 1 \cdot x^{2} = 36 x^{2}

36 \cdot 1 \cdot x^{2}

Multiplier les nombres :

36 \cdot 1 = 36

= 36 x^{2}

36 \cdot 108 x^{2} x = 3888 x^{3}

36 \cdot 108 x^{2} x

Multiplier les nombres :

36 \cdot 108 = 3888

= 3888 x^{2} x

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x = x^{2 + 1}

= 3888 x^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

= 36 x^{2} - 3888 x^{3}

Développer

(- 1 - 6 3 x 1 - (6 x)^{2})^{2} : 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

(- 1 - 6 3 x 1 - (6 x)^{2})^{2}

Appliquer la formule du carré parfait:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = - 1, b = 6 3 x 1 - (6 x)^{2}

= (- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

Simplifier

(- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2} : 1 + 12 3 x 1 - (6 x)^{2} + 363 x 1 - (6 x)^{2}

(- 1)^{2} - 2 (- 1) \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

Appliquer la règle

- (- a) = a

= (- 1)^{2} + 2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2} + (6 3 x 1 - (6 x)^{2})^{2}

(- 1)^{2} = 1

(- 1)^{2}

Appliquer la règle de l'exposant:

(- a)^{n} = a^{n},

n

pair

(- 1)^{2} = 1^{2}

= 1^{2}

Appliquer la règle

1^{a} = 1

= 1

2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2} = 12 3 x 1 - (6 x)^{2}

2 \cdot 1 \cdot 6 3 x 1 - (6 x)^{2}

Multiplier les nombres :

2 \cdot 1 \cdot 6 = 12

= 12 3 x 1 - (6 x)^{2}

(6 3 x 1 - (6 x)^{2})^{2} = 363 x 1 - (6 x)^{2}

(6 3 x 1 - (6 x)^{2})^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} (3 x)^{2} (1 - (6 x)^{2})^{2}

(3 x)^{2} : 3 x

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((3 x)^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= (3 x)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 3 x

= 6^{2} \cdot 3 x (1 - (6 x)^{2})^{2}

(1 - (6 x)^{2})^{2} : 1 - (6 x)^{2}

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((1 - (6 x)^{2})^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= (1 - (6 x)^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 1 - (6 x)^{2}

= 6^{2} \cdot 3 x (1 - (6 x)^{2})

6^{2} = 36

= 36 \cdot 3 x (1 - (6 x)^{2})

= 1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

= 1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

Développer

1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2}) : 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

1 + 12 3 x 1 - (6 x)^{2} + 36 \cdot 3 x (1 - (6 x)^{2})

12 3 x 1 - (6 x)^{2} = 123 x 1 - 36 x^{2}

12 3 x 1 - (6 x)^{2}

3 x = 3 x

3 x

Appliquer la règle des radicaux :

n ab = n a n b,

en supposant

a \geq 0, b \geq 0

= 3 x

= 123 x - (6 x)^{2} + 1

1 - (6 x)^{2} = 1 - 36 x^{2}

1 - (6 x)^{2}

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 - 36 x^{2}

= 123 x - 36 x^{2} + 1

36 \cdot 3 x (1 - (6 x)^{2}) = 108 x (1 - 36 x^{2})

36 \cdot 3 x (1 - (6 x)^{2})

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 36 \cdot 3 x (- 36 x^{2} + 1)

Multiplier les nombres :

36 \cdot 3 = 108

= 108 x (- 36 x^{2} + 1)

= 1 + 123 x - 36 x^{2} + 1 + 108 x (- 36 x^{2} + 1)

Développer

108 x (1 - 36 x^{2}) : 108 x - 3888 x^{3}

108 x (1 - 36 x^{2})

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = 108 x, b = 1, c = 36 x^{2}

= 108 x \cdot 1 - 108 x \cdot 36 x^{2}

= 108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x

Simplifier

108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x : 108 x - 3888 x^{3}

108 \cdot 1 \cdot x - 108 \cdot 36 x^{2} x

108 \cdot 1 \cdot x = 108 x

108 \cdot 1 \cdot x

Multiplier les nombres :

108 \cdot 1 = 108

= 108 x

108 \cdot 36 x^{2} x = 3888 x^{3}

108 \cdot 36 x^{2} x

Multiplier les nombres :

108 \cdot 36 = 3888

= 3888 x^{2} x

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x = x^{2 + 1}

= 3888 x^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= 3888 x^{3}

= 108 x - 3888 x^{3}

= 108 x - 3888 x^{3}

= 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

= 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

Soustraire

108 x - 3888 x^{3}

des deux côtés

36 x^{2} - 3888 x^{3} - (108 x - 3888 x^{3}) = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3} - (108 x - 3888 x^{3})

Simplifier

36 x^{2} - 108 x = 123 x 1 - 36 x^{2} + 1

Soustraire

1

des deux côtés

36 x^{2} - 108 x - 1 = 123 x 1 - 36 x^{2} + 1 - 1

Simplifier

36 x^{2} - 108 x - 1 = 123 x 1 - 36 x^{2}

Mettre les deux côtés au carré:

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

36 x^{2} - 3888 x^{3} = 1 + 123 x 1 - 36 x^{2} + 108 x - 3888 x^{3}

(36 x^{2} - 108 x - 1)^{2} = (123 x 1 - 36 x^{2})^{2}

Développer

(36 x^{2} - 108 x - 1)^{2} : 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

(36 x^{2} - 108 x - 1)^{2}

(36 x^{2} - 108 x - 1)^{2} = (36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

= (36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

Développer

(36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1) : 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

(36 x^{2} - 108 x - 1) (36 x^{2} - 108 x - 1)

Distribuer des parenthèses

= 36 x^{2} \cdot 36 x^{2} + 36 x^{2} (- 108 x) + 36 x^{2} (- 1) + (- 108 x) \cdot 36 x^{2} + (- 108 x) (- 108 x) + (- 108 x) (- 1) + (- 1) \cdot 36 x^{2} + (- 1) (- 108 x) + (- 1) (- 1)

Appliquer les règles des moins et des plus

+ (- a) = - a, (- a) (- b) = ab

= 36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

Simplifier

36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1 : 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

36 \cdot 36 x^{2} x^{2} - 36 \cdot 108 x^{2} x - 36 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

Additionner les éléments similaires :

- 36 \cdot 108 x^{2} x - 108 \cdot 36 x^{2} x = - 2 \cdot 108 \cdot 36 x^{2} x

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 36 \cdot 1 \cdot x^{2} + 108 \cdot 108 xx + 108 \cdot 1 \cdot x - 1 \cdot 36 x^{2} + 1 \cdot 108 x + 1 \cdot 1

Additionner les éléments similaires :

108 \cdot 1 \cdot x + 1 \cdot 108 x = 2 \cdot 1 \cdot 108 x

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 36 \cdot 1 \cdot x^{2} + 108 \cdot 108 xx + 2 \cdot 1 \cdot 108 x - 1 \cdot 36 x^{2} + 1 \cdot 1

Additionner les éléments similaires :

- 36 \cdot 1 \cdot x^{2} - 1 \cdot 36 x^{2} = - 2 \cdot 1 \cdot 36 x^{2}

= 36 \cdot 36 x^{2} x^{2} - 2 \cdot 108 \cdot 36 x^{2} x - 2 \cdot 1 \cdot 36 x^{2} + 108 \cdot 108 xx + 2 \cdot 1 \cdot 108 x + 1 \cdot 1

36 \cdot 36 x^{2} x^{2} = 1296 x^{4}

36 \cdot 36 x^{2} x^{2}

Multiplier les nombres :

36 \cdot 36 = 1296

= 1296 x^{2} x^{2}

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 1296 x^{2 + 2}

Additionner les nombres :

2 + 2 = 4

= 1296 x^{4}

2 \cdot 108 \cdot 36 x^{2} x = 7776 x^{3}

2 \cdot 108 \cdot 36 x^{2} x

Multiplier les nombres :

2 \cdot 108 \cdot 36 = 7776

= 7776 x^{2} x

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x = x^{2 + 1}

= 7776 x^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= 7776 x^{3}

2 \cdot 1 \cdot 36 x^{2} = 72 x^{2}

2 \cdot 1 \cdot 36 x^{2}

Multiplier les nombres :

2 \cdot 1 \cdot 36 = 72

= 72 x^{2}

108 \cdot 108 xx = 11664 x^{2}

108 \cdot 108 xx

Multiplier les nombres :

108 \cdot 108 = 11664

= 11664 xx

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= 11664 x^{1 + 1}

Additionner les nombres :

1 + 1 = 2

= 11664 x^{2}

2 \cdot 1 \cdot 108 x = 216 x

2 \cdot 1 \cdot 108 x

Multiplier les nombres :

2 \cdot 1 \cdot 108 = 216

= 216 x

1 \cdot 1 = 1

1 \cdot 1

Multiplier les nombres :

1 \cdot 1 = 1

= 1

= 1296 x^{4} - 7776 x^{3} - 72 x^{2} + 11664 x^{2} + 216 x + 1

Additionner les éléments similaires :

- 72 x^{2} + 11664 x^{2} = 11592 x^{2}

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

= 1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1

Développer

(123 x 1 - 36 x^{2})^{2} : 432 x - 15552 x^{3}

(123 x 1 - 36 x^{2})^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 1 2^{2} (3)^{2} (x)^{2} (1 - 36 x^{2})^{2}

(3)^{2} : 3

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 3

= 1 2^{2} \cdot 3 (x)^{2} (1 - 36 x^{2})^{2}

(x)^{2} : x

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (x^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= x^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= x

= 1 2^{2} \cdot 3 x (1 - 36 x^{2})^{2}

(1 - 36 x^{2})^{2} : 1 - 36 x^{2}

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((1 - 36 x^{2})^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= (1 - 36 x^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 1 - 36 x^{2}

= 1 2^{2} \cdot 3 x (1 - 36 x^{2})

Redéfinir

= 432 x (1 - 36 x^{2})

Développer

432 x (1 - 36 x^{2}) : 432 x - 15552 x^{3}

432 x (1 - 36 x^{2})

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = 432 x, b = 1, c = 36 x^{2}

= 432 x \cdot 1 - 432 x \cdot 36 x^{2}

= 432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x

Simplifier

432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x : 432 x - 15552 x^{3}

432 \cdot 1 \cdot x - 432 \cdot 36 x^{2} x

432 \cdot 1 \cdot x = 432 x

432 \cdot 1 \cdot x

Multiplier les nombres :

432 \cdot 1 = 432

= 432 x

432 \cdot 36 x^{2} x = 15552 x^{3}

432 \cdot 36 x^{2} x

Multiplier les nombres :

432 \cdot 36 = 15552

= 15552 x^{2} x

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x = x^{2 + 1}

= 15552 x^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= 15552 x^{3}

= 432 x - 15552 x^{3}

= 432 x - 15552 x^{3}

= 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

Résoudre

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3} : x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

Déplacer

15552 x^{3}

vers la gauche

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x - 15552 x^{3}

Ajouter

15552 x^{3}

aux deux côtés

1296 x^{4} - 7776 x^{3} + 11592 x^{2} + 216 x + 1 + 15552 x^{3} = 432 x - 15552 x^{3} + 15552 x^{3}

Simplifier

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

Déplacer

432 x

vers la gauche

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 = 432 x

Soustraire

432 x

des deux côtés

1296 x^{4} + 7776 x^{3} + 11592 x^{2} + 216 x + 1 - 432 x = 432 x - 432 x

Simplifier

1296 x^{4} + 7776 x^{3} + 11592 x^{2} - 216 x + 1 = 0

1296 x^{4} + 7776 x^{3} + 11592 x^{2} - 216 x + 1 = 0

Diviser les deux côtés par

1296

\frac{1296 x ^{4}}{1296} + \frac{7776 x ^{3}}{1296} + \frac{11592 x ^{2}}{1296} - \frac{216 x}{1296} + \frac{1}{1296} = \frac{0}{1296}

Ecrire sous la forme standard

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

x^{4} + 6 x^{3} + \frac{161 x ^{2}}{18} - \frac{x}{6} + \frac{1}{1296} = 0

Trouver une solution pour

x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots = 0

par la méthode de Newton-Raphson:

x \approx 0.00923 \dots

x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots = 0

Définition de l'approximation de Newton-Raphson

f (x) = x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots

Trouver

f^{^{'}} (x) : 4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots

\frac{d}{d x} (x^{4} + 6 x^{3} + 8.94444 \dots x^{2} - 0.16666 \dots x + 0.00077 \dots)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x^{4}) + \frac{d}{d x} (6 x^{3}) + \frac{d}{d x} (8.94444 \dots x^{2}) - \frac{d}{d x} (0.16666 \dots x) + \frac{d}{d x} (0.00077 \dots)

\frac{d}{d x} (x^{4}) = 4 x^{3}

\frac{d}{d x} (x^{4})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 x^{4 - 1}

Simplifier

= 4 x^{3}

\frac{d}{d x} (6 x^{3}) = 18 x^{2}

\frac{d}{d x} (6 x^{3})

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d}{d x} (x^{3})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 \cdot 3 x^{3 - 1}

Simplifier

= 18 x^{2}

\frac{d}{d x} (8.94444 \dots x^{2}) = 17.88888 \dots x

\frac{d}{d x} (8.94444 \dots x^{2})

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 8.94444 \dots \frac{d}{d x} (x^{2})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 8.94444 \dots \cdot 2 x^{2 - 1}

Simplifier

= 17.88888 \dots x

\frac{d}{d x} (0.16666 \dots x) = 0.16666 \dots

\frac{d}{d x} (0.16666 \dots x)

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.16666 \dots \frac{d x}{d x}

Appliquer la dérivée commune:

\frac{d x}{d x} = 1

= 0.16666 \dots \cdot 1

Simplifier

= 0.16666 \dots

\frac{d}{d x} (0.00077 \dots) = 0

\frac{d}{d x} (0.00077 \dots)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= 4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots + 0

Simplifier

= 4 x^{3} + 18 x^{2} + 17.88888 \dots x - 0.16666 \dots

Soit

x_{0} = 0

Calculer

x_{n + 1}

jusqu'à

Δ x_{n + 1} < 0.000001

x_{1} = 0.00462 \dots : Δ x_{1} = 0.00462 \dots

f (x_{0}) = 0^{4} + 6 \cdot 0^{3} + 8.94444 \dots \cdot 0^{2} - 0.16666 \dots \cdot 0 + 0.00077 \dots = 0.00077 \dots

f^{^{'}} (x_{0}) = 4 \cdot 0^{3} + 18 \cdot 0^{2} + 17.88888 \dots \cdot 0 - 0.16666 \dots = - 0.16666 \dots

x_{1} = 0.00462 \dots

Δ x_{1} = ∣ 0.00462 \dots - 0 ∣ = 0.00462 \dots

Δ x_{1} = 0.00462 \dots

x_{2} = 0.00693 \dots : Δ x_{2} = 0.00230 \dots

f (x_{1}) = 0.00462 \dots^{4} + 6 \cdot 0.00462 \dots^{3} + 8.94444 \dots \cdot 0.00462 \dots^{2} - 0.16666 \dots \cdot 0.00462 \dots + 0.00077 \dots = 0.00019 \dots

f^{^{'}} (x_{1}) = 4 \cdot 0.00462 \dots^{3} + 18 \cdot 0.00462 \dots^{2} + 17.88888 \dots \cdot 0.00462 \dots - 0.16666 \dots = - 0.08346 \dots

x_{2} = 0.00693 \dots

Δ x_{2} = ∣ 0.00693 \dots - 0.00462 \dots ∣ = 0.00230 \dots

Δ x_{2} = 0.00230 \dots

x_{3} = 0.00808 \dots : Δ x_{3} = 0.00114 \dots

f (x_{2}) = 0.00693 \dots^{4} + 6 \cdot 0.00693 \dots^{3} + 8.94444 \dots \cdot 0.00693 \dots^{2} - 0.16666 \dots \cdot 0.00693 \dots + 0.00077 \dots = 0.00004 \dots

f^{^{'}} (x_{2}) = 4 \cdot 0.00693 \dots^{3} + 18 \cdot 0.00693 \dots^{2} + 17.88888 \dots \cdot 0.00693 \dots - 0.16666 \dots = - 0.04176 \dots

x_{3} = 0.00808 \dots

Δ x_{3} = ∣ 0.00808 \dots - 0.00693 \dots ∣ = 0.00114 \dots

Δ x_{3} = 0.00114 \dots

x_{4} = 0.00865 \dots : Δ x_{4} = 0.00057 \dots

f (x_{3}) = 0.00808 \dots^{4} + 6 \cdot 0.00808 \dots^{3} + 8.94444 \dots \cdot 0.00808 \dots^{2} - 0.16666 \dots \cdot 0.00808 \dots + 0.00077 \dots = 0.00001 \dots

f^{^{'}} (x_{3}) = 4 \cdot 0.00808 \dots^{3} + 18 \cdot 0.00808 \dots^{2} + 17.88888 \dots \cdot 0.00808 \dots - 0.16666 \dots = - 0.02088 \dots

x_{4} = 0.00865 \dots

Δ x_{4} = ∣ 0.00865 \dots - 0.00808 \dots ∣ = 0.00057 \dots

Δ x_{4} = 0.00057 \dots

x_{5} = 0.00894 \dots : Δ x_{5} = 0.00028 \dots

f (x_{4}) = 0.00865 \dots^{4} + 6 \cdot 0.00865 \dots^{3} + 8.94444 \dots \cdot 0.00865 \dots^{2} - 0.16666 \dots \cdot 0.00865 \dots + 0.00077 \dots = 2.99676 E - 6

f^{^{'}} (x_{4}) = 4 \cdot 0.00865 \dots^{3} + 18 \cdot 0.00865 \dots^{2} + 17.88888 \dots \cdot 0.00865 \dots - 0.16666 \dots = - 0.01044 \dots

x_{5} = 0.00894 \dots

Δ x_{5} = ∣ 0.00894 \dots - 0.00865 \dots ∣ = 0.00028 \dots

Δ x_{5} = 0.00028 \dots

x_{6} = 0.00908 \dots : Δ x_{6} = 0.00014 \dots

f (x_{5}) = 0.00894 \dots^{4} + 6 \cdot 0.00894 \dots^{3} + 8.94444 \dots \cdot 0.00894 \dots^{2} - 0.16666 \dots \cdot 0.00894 \dots + 0.00077 \dots = 7.49047 E - 7

f^{^{'}} (x_{5}) = 4 \cdot 0.00894 \dots^{3} + 18 \cdot 0.00894 \dots^{2} + 17.88888 \dots \cdot 0.00894 \dots - 0.16666 \dots = - 0.00522 \dots

x_{6} = 0.00908 \dots

Δ x_{6} = ∣ 0.00908 \dots - 0.00894 \dots ∣ = 0.00014 \dots

Δ x_{6} = 0.00014 \dots

x_{7} = 0.00915 \dots : Δ x_{7} = 0.00007 \dots

f (x_{6}) = 0.00908 \dots^{4} + 6 \cdot 0.00908 \dots^{3} + 8.94444 \dots \cdot 0.00908 \dots^{2} - 0.16666 \dots \cdot 0.00908 \dots + 0.00077 \dots = 1.87244 E - 7

f^{^{'}} (x_{6}) = 4 \cdot 0.00908 \dots^{3} + 18 \cdot 0.00908 \dots^{2} + 17.88888 \dots \cdot 0.00908 \dots - 0.16666 \dots = - 0.00261 \dots

x_{7} = 0.00915 \dots

Δ x_{7} = ∣ 0.00915 \dots - 0.00908 \dots ∣ = 0.00007 \dots

Δ x_{7} = 0.00007 \dots

x_{8} = 0.00919 \dots : Δ x_{8} = 0.00003 \dots

f (x_{7}) = 0.00915 \dots^{4} + 6 \cdot 0.00915 \dots^{3} + 8.94444 \dots \cdot 0.00915 \dots^{2} - 0.16666 \dots \cdot 0.00915 \dots + 0.00077 \dots = 4.68088 E - 8

f^{^{'}} (x_{7}) = 4 \cdot 0.00915 \dots^{3} + 18 \cdot 0.00915 \dots^{2} + 17.88888 \dots \cdot 0.00915 \dots - 0.16666 \dots = - 0.00130 \dots

x_{8} = 0.00919 \dots

Δ x_{8} = ∣ 0.00919 \dots - 0.00915 \dots ∣ = 0.00003 \dots

Δ x_{8} = 0.00003 \dots

x_{9} = 0.00921 \dots : Δ x_{9} = 0.00001 \dots

f (x_{8}) = 0.00919 \dots^{4} + 6 \cdot 0.00919 \dots^{3} + 8.94444 \dots \cdot 0.00919 \dots^{2} - 0.16666 \dots \cdot 0.00919 \dots + 0.00077 \dots = 1.17019 E - 8

f^{^{'}} (x_{8}) = 4 \cdot 0.00919 \dots^{3} + 18 \cdot 0.00919 \dots^{2} + 17.88888 \dots \cdot 0.00919 \dots - 0.16666 \dots = - 0.00065 \dots

x_{9} = 0.00921 \dots

Δ x_{9} = ∣ 0.00921 \dots - 0.00919 \dots ∣ = 0.00001 \dots

Δ x_{9} = 0.00001 \dots

x_{10} = 0.00922 \dots : Δ x_{10} = 8.95953 E - 6

f (x_{9}) = 0.00921 \dots^{4} + 6 \cdot 0.00921 \dots^{3} + 8.94444 \dots \cdot 0.00921 \dots^{2} - 0.16666 \dots \cdot 0.00921 \dots + 0.00077 \dots = 2.92544 E - 9

f^{^{'}} (x_{9}) = 4 \cdot 0.00921 \dots^{3} + 18 \cdot 0.00921 \dots^{2} + 17.88888 \dots \cdot 0.00921 \dots - 0.16666 \dots = - 0.00032 \dots

x_{10} = 0.00922 \dots

Δ x_{10} = ∣ 0.00922 \dots - 0.00921 \dots ∣ = 8.95953 E - 6

Δ x_{10} = 8.95953 E - 6

x_{11} = 0.00922 \dots : Δ x_{11} = 4.47973 E - 6

f (x_{10}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 7.31357 E - 10

f^{^{'}} (x_{10}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00016 \dots

x_{11} = 0.00922 \dots

Δ x_{11} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 4.47973 E - 6

Δ x_{11} = 4.47973 E - 6

x_{12} = 0.00922 \dots : Δ x_{12} = 2.23985 E - 6

f (x_{11}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 1.82839 E - 10

f^{^{'}} (x_{11}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00008 \dots

x_{12} = 0.00922 \dots

Δ x_{12} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 2.23985 E - 6

Δ x_{12} = 2.23985 E - 6

x_{13} = 0.00922 \dots : Δ x_{13} = 1.11992 E - 6

f (x_{12}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 4.57096 E - 11

f^{^{'}} (x_{12}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00004 \dots

x_{13} = 0.00922 \dots

Δ x_{13} = ∣ 0.00922 \dots - 0.00922 \dots ∣ = 1.11992 E - 6

Δ x_{13} = 1.11992 E - 6

x_{14} = 0.00923 \dots : Δ x_{14} = 5.59961 E - 7

f (x_{13}) = 0.00922 \dots^{4} + 6 \cdot 0.00922 \dots^{3} + 8.94444 \dots \cdot 0.00922 \dots^{2} - 0.16666 \dots \cdot 0.00922 \dots + 0.00077 \dots = 1.14274 E - 11

f^{^{'}} (x_{13}) = 4 \cdot 0.00922 \dots^{3} + 18 \cdot 0.00922 \dots^{2} + 17.88888 \dots \cdot 0.00922 \dots - 0.16666 \dots = - 0.00002 \dots

x_{14} = 0.00923 \dots

Δ x_{14} = ∣ 0.00923 \dots - 0.00922 \dots ∣ = 5.59961 E - 7

Δ x_{14} = 5.59961 E - 7

x \approx 0.00923 \dots

Appliquer une division longue:

\frac{x ^{4} + 6 x ^{3} + \frac{161 x ^{2}}{18} - \frac{x}{6} + \frac{1}{1296}}{x - 0.00923 \dots} = x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots \approx 0

Trouver une solution pour

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots = 0

par la méthode de Newton-Raphson:

x \approx 0.00923 \dots

x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots = 0

Définition de l'approximation de Newton-Raphson

f (x) = x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots

Trouver

f^{^{'}} (x) : 3 x^{2} + 12.01846 \dots x + 8.99991 \dots

\frac{d}{d x} (x^{3} + 6.00923 \dots x^{2} + 8.99991 \dots x - 0.08359 \dots)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x^{3}) + \frac{d}{d x} (6.00923 \dots x^{2}) + \frac{d}{d x} (8.99991 \dots x) - \frac{d}{d x} (0.08359 \dots)

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} (x^{3})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 x^{3 - 1}

Simplifier

= 3 x^{2}

\frac{d}{d x} (6.00923 \dots x^{2}) = 12.01846 \dots x

\frac{d}{d x} (6.00923 \dots x^{2})

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 6.00923 \dots \frac{d}{d x} (x^{2})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6.00923 \dots \cdot 2 x^{2 - 1}

Simplifier

= 12.01846 \dots x

\frac{d}{d x} (8.99991 \dots x) = 8.99991 \dots

\frac{d}{d x} (8.99991 \dots x)

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 8.99991 \dots \frac{d x}{d x}

Appliquer la dérivée commune:

\frac{d x}{d x} = 1

= 8.99991 \dots \cdot 1

Simplifier

= 8.99991 \dots

\frac{d}{d x} (0.08359 \dots) = 0

\frac{d}{d x} (0.08359 \dots)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= 3 x^{2} + 12.01846 \dots x + 8.99991 \dots - 0

Simplifier

= 3 x^{2} + 12.01846 \dots x + 8.99991 \dots

Soit

x_{0} = 0

Calculer

x_{n + 1}

jusqu'à

Δ x_{n + 1} < 0.000001

x_{1} = 0.00928 \dots : Δ x_{1} = 0.00928 \dots

f (x_{0}) = 0^{3} + 6.00923 \dots \cdot 0^{2} + 8.99991 \dots \cdot 0 - 0.08359 \dots = - 0.08359 \dots

f^{^{'}} (x_{0}) = 3 \cdot 0^{2} + 12.01846 \dots \cdot 0 + 8.99991 \dots = 8.99991 \dots

x_{1} = 0.00928 \dots

Δ x_{1} = ∣ 0.00928 \dots - 0 ∣ = 0.00928 \dots

Δ x_{1} = 0.00928 \dots

x_{2} = 0.00923 \dots : Δ x_{2} = 0.00005 \dots

f (x_{1}) = 0.00928 \dots^{3} + 6.00923 \dots \cdot 0.00928 \dots^{2} + 8.99991 \dots \cdot 0.00928 \dots - 0.08359 \dots = 0.00051 \dots

f^{^{'}} (x_{1}) = 3 \cdot 0.00928 \dots^{2} + 12.01846 \dots \cdot 0.00928 \dots + 8.99991 \dots = 9.11180 \dots

x_{2} = 0.00923 \dots

Δ x_{2} = ∣ 0.00923 \dots - 0.00928 \dots ∣ = 0.00005 \dots

Δ x_{2} = 0.00005 \dots

x_{3} = 0.00923 \dots : Δ x_{3} = 2.15173 E - 9

f (x_{2}) = 0.00923 \dots^{3} + 6.00923 \dots \cdot 0.00923 \dots^{2} + 8.99991 \dots \cdot 0.00923 \dots - 0.08359 \dots = 1.96046 E - 8

f^{^{'}} (x_{2}) = 3 \cdot 0.00923 \dots^{2} + 12.01846 \dots \cdot 0.00923 \dots + 8.99991 \dots = 9.11111 \dots

x_{3} = 0.00923 \dots

Δ x_{3} = ∣ 0.00923 \dots - 0.00923 \dots ∣ = 2.15173 E - 9

Δ x_{3} = 2.15173 E - 9

x \approx 0.00923 \dots

Appliquer une division longue:

\frac{x ^{3} + 6.00923 \dots x ^{2} + 8.99991 \dots x - 0.08359 \dots}{x - 0.00923 \dots} = x^{2} + 6.01846 \dots x + 9.05547 \dots

x^{2} + 6.01846 \dots x + 9.05547 \dots \approx 0

Trouver une solution pour

x^{2} + 6.01846 \dots x + 9.05547 \dots = 0

par la méthode de Newton-Raphson:

x \approx - 3.00922 \dots

x^{2} + 6.01846 \dots x + 9.05547 \dots = 0

Définition de l'approximation de Newton-Raphson

f (x) = x^{2} + 6.01846 \dots x + 9.05547 \dots

Trouver

f^{^{'}} (x) : 2 x + 6.01846 \dots

\frac{d}{d x} (x^{2} + 6.01846 \dots x + 9.05547 \dots)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6.01846 \dots x) + \frac{d}{d x} (9.05547 \dots)

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (x^{2})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 x^{2 - 1}

Simplifier

= 2 x

\frac{d}{d x} (6.01846 \dots x) = 6.01846 \dots

\frac{d}{d x} (6.01846 \dots x)

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 6.01846 \dots \frac{d x}{d x}

Appliquer la dérivée commune:

\frac{d x}{d x} = 1

= 6.01846 \dots \cdot 1

Simplifier

= 6.01846 \dots

\frac{d}{d x} (9.05547 \dots) = 0

\frac{d}{d x} (9.05547 \dots)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= 2 x + 6.01846 \dots + 0

Simplifier

= 2 x + 6.01846 \dots

Soit

x_{0} = - 2

Calculer

x_{n + 1}

jusqu'à

Δ x_{n + 1} < 0.000001

x_{1} = - 2.50461 \dots : Δ x_{1} = 0.50461 \dots

f (x_{0}) = (- 2)^{2} + 6.01846 \dots (- 2) + 9.05547 \dots = 1.01854 \dots

f^{^{'}} (x_{0}) = 2 (- 2) + 6.01846 \dots = 2.01846 \dots

x_{1} = - 2.50461 \dots

Δ x_{1} = ∣ - 2.50461 \dots - (- 2) ∣ = 0.50461 \dots

Δ x_{1} = 0.50461 \dots

x_{2} = - 2.75692 \dots : Δ x_{2} = 0.25230 \dots

f (x_{1}) = (- 2.50461 \dots)^{2} + 6.01846 \dots (- 2.50461 \dots) + 9.05547 \dots = 0.25463 \dots

f^{^{'}} (x_{1}) = 2 (- 2.50461 \dots) + 6.01846 \dots = 1.00923 \dots

x_{2} = - 2.75692 \dots

Δ x_{2} = ∣ - 2.75692 \dots - (- 2.50461 \dots) ∣ = 0.25230 \dots

Δ x_{2} = 0.25230 \dots

x_{3} = - 2.88307 \dots : Δ x_{3} = 0.12615 \dots

f (x_{2}) = (- 2.75692 \dots)^{2} + 6.01846 \dots (- 2.75692 \dots) + 9.05547 \dots = 0.06365 \dots

f^{^{'}} (x_{2}) = 2 (- 2.75692 \dots) + 6.01846 \dots = 0.50461 \dots

x_{3} = - 2.88307 \dots

Δ x_{3} = ∣ - 2.88307 \dots - (- 2.75692 \dots) ∣ = 0.12615 \dots

Δ x_{3} = 0.12615 \dots

x_{4} = - 2.94615 \dots : Δ x_{4} = 0.06307 \dots

f (x_{3}) = (- 2.88307 \dots)^{2} + 6.01846 \dots (- 2.88307 \dots) + 9.05547 \dots = 0.01591 \dots

f^{^{'}} (x_{3}) = 2 (- 2.88307 \dots) + 6.01846 \dots = 0.25230 \dots

x_{4} = - 2.94615 \dots

Δ x_{4} = ∣ - 2.94615 \dots - (- 2.88307 \dots) ∣ = 0.06307 \dots

Δ x_{4} = 0.06307 \dots

x_{5} = - 2.97769 \dots : Δ x_{5} = 0.03153 \dots

f (x_{4}) = (- 2.94615 \dots)^{2} + 6.01846 \dots (- 2.94615 \dots) + 9.05547 \dots = 0.00397 \dots

f^{^{'}} (x_{4}) = 2 (- 2.94615 \dots) + 6.01846 \dots = 0.12615 \dots

x_{5} = - 2.97769 \dots

Δ x_{5} = ∣ - 2.97769 \dots - (- 2.94615 \dots) ∣ = 0.03153 \dots

Δ x_{5} = 0.03153 \dots

x_{6} = - 2.99346 \dots : Δ x_{6} = 0.01576 \dots

f (x_{5}) = (- 2.97769 \dots)^{2} + 6.01846 \dots (- 2.97769 \dots) + 9.05547 \dots = 0.00099 \dots

f^{^{'}} (x_{5}) = 2 (- 2.97769 \dots) + 6.01846 \dots = 0.06307 \dots

x_{6} = - 2.99346 \dots

Δ x_{6} = ∣ - 2.99346 \dots - (- 2.97769 \dots) ∣ = 0.01576 \dots

Δ x_{6} = 0.01576 \dots

x_{7} = - 3.00134 \dots : Δ x_{7} = 0.00788 \dots

f (x_{6}) = (- 2.99346 \dots)^{2} + 6.01846 \dots (- 2.99346 \dots) + 9.05547 \dots = 0.00024 \dots

f^{^{'}} (x_{6}) = 2 (- 2.99346 \dots) + 6.01846 \dots = 0.03153 \dots

x_{7} = - 3.00134 \dots

Δ x_{7} = ∣ - 3.00134 \dots - (- 2.99346 \dots) ∣ = 0.00788 \dots

Δ x_{7} = 0.00788 \dots

x_{8} = - 3.00528 \dots : Δ x_{8} = 0.00394 \dots

f (x_{7}) = (- 3.00134 \dots)^{2} + 6.01846 \dots (- 3.00134 \dots) + 9.05547 \dots = 0.00006 \dots

f^{^{'}} (x_{7}) = 2 (- 3.00134 \dots) + 6.01846 \dots = 0.01576 \dots

x_{8} = - 3.00528 \dots

Δ x_{8} = ∣ - 3.00528 \dots - (- 3.00134 \dots) ∣ = 0.00394 \dots

Δ x_{8} = 0.00394 \dots

x_{9} = - 3.00725 \dots : Δ x_{9} = 0.00197 \dots

f (x_{8}) = (- 3.00528 \dots)^{2} + 6.01846 \dots (- 3.00528 \dots) + 9.05547 \dots = 0.00001 \dots

f^{^{'}} (x_{8}) = 2 (- 3.00528 \dots) + 6.01846 \dots = 0.00788 \dots

x_{9} = - 3.00725 \dots

Δ x_{9} = ∣ - 3.00725 \dots - (- 3.00528 \dots) ∣ = 0.00197 \dots

Δ x_{9} = 0.00197 \dots

x_{10} = - 3.00824 \dots : Δ x_{10} = 0.00098 \dots

f (x_{9}) = (- 3.00725 \dots)^{2} + 6.01846 \dots (- 3.00725 \dots) + 9.05547 \dots = 3.88545 E - 6

f^{^{'}} (x_{9}) = 2 (- 3.00725 \dots) + 6.01846 \dots = 0.00394 \dots

x_{10} = - 3.00824 \dots

Δ x_{10} = ∣ - 3.00824 \dots - (- 3.00725 \dots) ∣ = 0.00098 \dots

Δ x_{10} = 0.00098 \dots

x_{11} = - 3.00873 \dots : Δ x_{11} = 0.00049 \dots

f (x_{10}) = (- 3.00824 \dots)^{2} + 6.01846 \dots (- 3.00824 \dots) + 9.05547 \dots = 9.71362 E - 7

f^{^{'}} (x_{10}) = 2 (- 3.00824 \dots) + 6.01846 \dots = 0.00197 \dots

x_{11} = - 3.00873 \dots

Δ x_{11} = ∣ - 3.00873 \dots - (- 3.00824 \dots) ∣ = 0.00049 \dots

Δ x_{11} = 0.00049 \dots

x_{12} = - 3.00898 \dots : Δ x_{12} = 0.00024 \dots

f (x_{11}) = (- 3.00873 \dots)^{2} + 6.01846 \dots (- 3.00873 \dots) + 9.05547 \dots = 2.4284 E - 7

f^{^{'}} (x_{11}) = 2 (- 3.00873 \dots) + 6.01846 \dots = 0.00098 \dots

x_{12} = - 3.00898 \dots

Δ x_{12} = ∣ - 3.00898 \dots - (- 3.00873 \dots) ∣ = 0.00024 \dots

Δ x_{12} = 0.00024 \dots

x_{13} = - 3.00910 \dots : Δ x_{13} = 0.00012 \dots

f (x_{12}) = (- 3.00898 \dots)^{2} + 6.01846 \dots (- 3.00898 \dots) + 9.05547 \dots = 6.071 E - 8

f^{^{'}} (x_{12}) = 2 (- 3.00898 \dots) + 6.01846 \dots = 0.00049 \dots

x_{13} = - 3.00910 \dots

Δ x_{13} = ∣ - 3.00910 \dots - (- 3.00898 \dots) ∣ = 0.00012 \dots

Δ x_{13} = 0.00012 \dots

x_{14} = - 3.00916 \dots : Δ x_{14} = 0.00006 \dots

f (x_{13}) = (- 3.00910 \dots)^{2} + 6.01846 \dots (- 3.00910 \dots) + 9.05547 \dots = 1.51774 E - 8

f^{^{'}} (x_{13}) = 2 (- 3.00910 \dots) + 6.01846 \dots = 0.00024 \dots

x_{14} = - 3.00916 \dots

Δ x_{14} = ∣ - 3.00916 \dots - (- 3.00910 \dots) ∣ = 0.00006 \dots

Δ x_{14} = 0.00006 \dots

x_{15} = - 3.00920 \dots : Δ x_{15} = 0.00003 \dots

f (x_{14}) = (- 3.00916 \dots)^{2} + 6.01846 \dots (- 3.00916 \dots) + 9.05547 \dots = 3.79428 E - 9

f^{^{'}} (x_{14}) = 2 (- 3.00916 \dots) + 6.01846 \dots = 0.00012 \dots

x_{15} = - 3.00920 \dots

Δ x_{15} = ∣ - 3.00920 \dots - (- 3.00916 \dots) ∣ = 0.00003 \dots

Δ x_{15} = 0.00003 \dots

x_{16} = - 3.00921 \dots : Δ x_{16} = 0.00001 \dots

f (x_{15}) = (- 3.00920 \dots)^{2} + 6.01846 \dots (- 3.00920 \dots) + 9.05547 \dots = 9.48491 E - 10

f^{^{'}} (x_{15}) = 2 (- 3.00920 \dots) + 6.01846 \dots = 0.00006 \dots

x_{16} = - 3.00921 \dots

Δ x_{16} = ∣ - 3.00921 \dots - (- 3.00920 \dots) ∣ = 0.00001 \dots

Δ x_{16} = 0.00001 \dots

x_{17} = - 3.00922 \dots : Δ x_{17} = 7.69303 E - 6

f (x_{16}) = (- 3.00921 \dots)^{2} + 6.01846 \dots (- 3.00921 \dots) + 9.05547 \dots = 2.37044 E - 10

f^{^{'}} (x_{16}) = 2 (- 3.00921 \dots) + 6.01846 \dots = 0.00003 \dots

x_{17} = - 3.00922 \dots

Δ x_{17} = ∣ - 3.00922 \dots - (- 3.00921 \dots) ∣ = 7.69303 E - 6

Δ x_{17} = 7.69303 E - 6

x_{18} = - 3.00922 \dots : Δ x_{18} = 3.83637 E - 6

f (x_{17}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 5.91829 E - 11

f^{^{'}} (x_{17}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 0.00001 \dots

x_{18} = - 3.00922 \dots

Δ x_{18} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 3.83637 E - 6

Δ x_{18} = 3.83637 E - 6

x_{19} = - 3.00922 \dots : Δ x_{19} = 1.89799 E - 6

f (x_{18}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 1.47171 E - 11

f^{^{'}} (x_{18}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 7.75406 E - 6

x_{19} = - 3.00922 \dots

Δ x_{19} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 1.89799 E - 6

Δ x_{19} = 1.89799 E - 6

x_{20} = - 3.00922 \dots : Δ x_{20} = 9.10151 E - 7

f (x_{19}) = (- 3.00922 \dots)^{2} + 6.01846 \dots (- 3.00922 \dots) + 9.05547 \dots = 3.60245 E - 12

f^{^{'}} (x_{19}) = 2 (- 3.00922 \dots) + 6.01846 \dots = 3.95808 E - 6

x_{20} = - 3.00922 \dots

Δ x_{20} = ∣ - 3.00922 \dots - (- 3.00922 \dots) ∣ = 9.10151 E - 7

Δ x_{20} = 9.10151 E - 7

x \approx - 3.00922 \dots

Appliquer une division longue:

\frac{x ^{2} + 6.01846 \dots x + 9.05547 \dots}{x + 3.00922 \dots} = x + 3.00923 \dots

x + 3.00923 \dots \approx 0

x \approx - 3.00923 \dots

Les solutions sont

x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

x \approx 0.00923 \dots, x \approx 0.00923 \dots, x \approx - 3.00922 \dots, x \approx - 3.00923 \dots

Vérifier les solutions:

x \approx 0.00923 \dots

Faux

, x \approx 0.00923 \dots

Faux

, x \approx - 3.00922 \dots

Faux

, x \approx - 3.00923 \dots

Faux

Vérifier des solutions en les intégrant dans

6 x 1 - (6 3 x)^{2} + 6 3 x 1 - (6 x)^{2} = - 1

Retirer celles qui ne répondent pas à l'équation.

Insérer

x \approx 0.00923 \dots :

Faux

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = - 1

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 0.99999 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.05538 \dots 0.00312 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2}

1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.00312 \dots

1 - (6 3 \cdot 0.00923 \dots)^{2}

(6 3 \cdot 0.00923 \dots)^{2} = 0.99687 \dots

(6 3 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

3 \cdot 0.00923 \dots = 0.02769 \dots

= (6 0.02769 \dots)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} (0.02769 \dots)^{2}

(0.02769 \dots)^{2} : 0.02769 \dots

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (0.02769 \dots^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= 0.02769 \dots^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 0.02769 \dots

= 6^{2} \cdot 0.02769 \dots

6^{2} = 36

= 36 \cdot 0.02769 \dots

Multiplier les nombres :

36 \cdot 0.02769 \dots = 0.99687 \dots

= 0.99687 \dots

= 1 - 0.99687 \dots

Soustraire les nombres :

1 - 0.99687 \dots = 0.00312 \dots

= 0.00312 \dots

= 6 \cdot 0.00923 \dots 0.00312 \dots

Multiplier les nombres :

6 \cdot 0.00923 \dots = 0.05538 \dots

= 0.05538 \dots 0.00312 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 6 0.02760 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

3 \cdot 0.00923 \dots = 0.02769 \dots

= 6 0.02769 \dots - (6 \cdot 0.00923 \dots)^{2} + 1

1 - (6 \cdot 0.00923 \dots)^{2} = 0.99693 \dots

1 - (6 \cdot 0.00923 \dots)^{2}

(6 \cdot 0.00923 \dots)^{2} = 0.00306 \dots

(6 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

6 \cdot 0.00923 \dots = 0.05538 \dots

= 0.05538 \dots^{2}

0.05538 \dots^{2} = 0.00306 \dots

= 0.00306 \dots

= 1 - 0.00306 \dots

Soustraire les nombres :

1 - 0.00306 \dots = 0.99693 \dots

= 0.99693 \dots

= 6 0.02769 \dots 0.99693 \dots

Appliquer la règle des radicaux:

a b = a \cdot b

0.02769 \dots 0.99693 \dots = 0.02769 \dots \cdot 0.99693 \dots

= 6 0.02769 \dots \cdot 0.99693 \dots

Multiplier les nombres :

0.02769 \dots \cdot 0.99693 \dots = 0.02760 \dots

= 6 0.02760 \dots

= 0.05538 \dots 0.00312 \dots + 6 0.02760 \dots

0.05538 \dots 0.00312 \dots = 0.00309 \dots

0.05538 \dots 0.00312 \dots

0.00312 \dots = 0.05592 \dots

= 0.05538 \dots \cdot 0.05592 \dots

Multiplier les nombres :

0.05538 \dots \cdot 0.05592 \dots = 0.00309 \dots

= 0.00309 \dots

6 0.02760 \dots = 0.99690 \dots

6 0.02760 \dots

0.02760 \dots = 0.16615 \dots

= 6 \cdot 0.16615 \dots

Multiplier les nombres :

6 \cdot 0.16615 \dots = 0.99690 \dots

= 0.99690 \dots

= 0.00309 \dots + 0.99690 \dots

Additionner les nombres :

0.00309 \dots + 0.99690 \dots = 0.99999 \dots

= 0.99999 \dots

0.99999 \dots = - 1

Faux

Insérer

x \approx 0.00923 \dots :

Faux

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = - 1

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 0.99999 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} + 6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.05538 \dots 0.00300 \dots

6 \cdot 0.00923 \dots 1 - (6 3 \cdot 0.00923 \dots)^{2}

1 - (6 3 \cdot 0.00923 \dots)^{2} = 0.00300 \dots

1 - (6 3 \cdot 0.00923 \dots)^{2}

(6 3 \cdot 0.00923 \dots)^{2} = 0.99699 \dots

(6 3 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

3 \cdot 0.00923 \dots = 0.02769 \dots

= (6 0.02769 \dots)^{2}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} (0.02769 \dots)^{2}

(0.02769 \dots)^{2} : 0.02769 \dots

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= (0.02769 \dots^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

(a^{b})^{c} = a^{b c}

= 0.02769 \dots^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 0.02769 \dots

= 6^{2} \cdot 0.02769 \dots

6^{2} = 36

= 36 \cdot 0.02769 \dots

Multiplier les nombres :

36 \cdot 0.02769 \dots = 0.99699 \dots

= 0.99699 \dots

= 1 - 0.99699 \dots

Soustraire les nombres :

1 - 0.99699 \dots = 0.00300 \dots

= 0.00300 \dots

= 6 \cdot 0.00923 \dots 0.00300 \dots

Multiplier les nombres :

6 \cdot 0.00923 \dots = 0.05538 \dots

= 0.05538 \dots 0.00300 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2} = 6 0.02760 \dots

6 3 \cdot 0.00923 \dots 1 - (6 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

3 \cdot 0.00923 \dots = 0.02769 \dots

= 6 0.02769 \dots - (6 \cdot 0.00923 \dots)^{2} + 1

1 - (6 \cdot 0.00923 \dots)^{2} = 0.99693 \dots

1 - (6 \cdot 0.00923 \dots)^{2}

(6 \cdot 0.00923 \dots)^{2} = 0.00306 \dots

(6 \cdot 0.00923 \dots)^{2}

Multiplier les nombres :

6 \cdot 0.00923 \dots = 0.05538 \dots

= 0.05538 \dots^{2}

0.05538 \dots^{2} = 0.00306 \dots

= 0.00306 \dots

= 1 - 0.00306 \dots

Soustraire les nombres :

1 - 0.00306 \dots = 0.99693 \dots

= 0.99693 \dots

= 6 0.02769 \dots 0.99693 \dots

Appliquer la règle des radicaux:

a b = a \cdot b

0.02769 \dots 0.99693 \dots = 0.02769 \dots \cdot 0.99693 \dots

= 6 0.02769 \dots \cdot 0.99693 \dots

Multiplier les nombres :

0.02769 \dots \cdot 0.99693 \dots = 0.02760 \dots

= 6 0.02760 \dots

= 0.05538 \dots 0.00300 \dots + 6 0.02760 \dots

0.05538 \dots 0.00300 \dots = 0.00303 \dots

0.05538 \dots 0.00300 \dots

0.00300 \dots = 0.05483 \dots

= 0.05538 \dots \cdot 0.05483 \dots

Multiplier les nombres :

0.05538 \dots \cdot 0.05483 \dots = 0.00303 \dots

= 0.00303 \dots

6 0.02760 \dots = 0.99696 \dots

6 0.02760 \dots

0.02760 \dots = 0.16616 \dots

= 6 \cdot 0.16616 \dots

Multiplier les nombres :

6 \cdot 0.16616 \dots = 0.99696 \dots

= 0.99696 \dots

= 0.00303 \dots + 0.99696 \dots

Additionner les nombres :

0.00303 \dots + 0.99696 \dots = 0.99999 \dots

= 0.99999 \dots

0.99999 \dots = - 1

Faux

Insérer

x \approx - 3.00922 \dots :

Faux

6 (- 3.00922 \dots) 1 - (6 3 (- 3.00922 \dots))^{2} + 6 3 (- 3.00922 \dots) 1 - (6 (- 3.00922 \dots))^{2} = - 1

6 (- 3.00922 \dots) 1 - (6 3 (- 3.00922 \dots))^{2} + 6 3 (- 3.00922 \dots) 1 - (6 (- 3.00922 \dots))^{2} =

Indéfini

= - 1

Faux

Insérer

x \approx - 3.00923 \dots :

Faux

6 (- 3.00923 \dots) 1 - (6 3 (- 3.00923 \dots))^{2} + 6 3 (- 3.00923 \dots) 1 - (6 (- 3.00923 \dots))^{2} = - 1

6 (- 3.00923 \dots) 1 - (6 3 (- 3.00923 \dots))^{2} + 6 3 (- 3.00923 \dots) 1 - (6 (- 3.00923 \dots))^{2} =

Indéfini

= - 1

Faux

La solution est

Aucune solution pour x \in R

Aucune solution

Vérifier les solutions en les intégrant dans l'équation d'origine

Vérifier des solutions en les intégrant dans

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}

Retirer celles qui ne répondent pas à l'équation.

Aucune solution pour x \in R

Graphe

Exemples populaires

2cos^2(x)+sin(x)=5

2 cos^{2} (x) + sin (x) = 5

solvefor x,Y=0.5sin(3.07x-2.4t+0.59)

so l v e f or x, Y = 0.5 sin (3.07 x - 2.4 t + 0.59)

sin(x-30)cos(x-30)=(sqrt(3))/4

sin (x - 3 0^{\circ}) cos (x - 3 0^{\circ}) = \frac{3}{4}

cos(θ)=-7/15 ,cos(θ/2),180<θ<270

cos (θ) = - \frac{7}{15}, cos (\frac{θ}{2}), 18 0^{\circ} < θ < 27 0^{\circ}

tan(x/2)=4

tan (\frac{x}{2}) = 4