Je me demande quelles seraient les solutions de l'équation x² + 4x + 4 = 0.
Quadratic equations

A quadratic equation is one which includes a term with an exponent of 2, for example, x 2 {\displaystyle x^{2}}, and no term with higher exponent. The name derives from the Latin quadrus, meaning square. In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0}, where a is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term a x 2 {\displaystyle ax^{2}}, which is known as the quadratic term. Hence a ≠ 0 {\displaystyle a\neq 0}, and so we may divide by a and rearrange the equation into the standard form

where p = b a {\displaystyle p={\frac {b}{a}}} and q = c a {\displaystyle q={\frac {c}{a}}}. Solving this, by a process known as completing the square, leads to the quadratic formula

where the symbol "±" indicates that both

are solutions of the quadratic equation.

Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:

which is the same thing as

It follows from the zero-product property that either x = 2 {\displaystyle x=2} or x = − 5 {\displaystyle x=-5} are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,

has no real number solution since no real number squared equals −1.Sometimes a quadratic equation has a root of multiplicity 2, such as:

For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
Équations quatriques

Pour des discussions détaillées de certaines méthodes de résolution, voir :

Une équation quatrique a x⁴ + b x³ + c x² + d x + e = 0 avec a ≠ 0 peut être réduite à une équation quadratique par un changement de variable à condition qu'elle soit soit biquadratique (b = d = 0) soit quasi-palindromique (e = a, d = b).

Certaines équations cubiques et quatriques peuvent être résolues en utilisant la trigonométrie ou les fonctions hyperboliques.
### Query ###
Je me demande quelles seraient les solutions de l'équation x² + 4x + 4 = 0.

### Main Text ###
Quadratic equations

A quadratic equation is one which includes a term with an exponent of 2, for example, x 2 {\displaystyle x^{2}}, and no term with higher exponent. The name derives from the Latin quadrus, meaning square. In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0}, where a is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term a x 2 {\displaystyle ax^{2}}, which is known as the quadratic term. Hence a ≠ 0 {\displaystyle a\neq 0}, and so we may divide by a and rearrange the equation into the standard form

where p = b a {\displaystyle p={\frac {b}{a}}} and q = c a {\displaystyle q={\frac {c}{a}}}. Solving this, by a process known as completing the square, leads to the quadratic formula

where the symbol "±" indicates that both

are solutions of the quadratic equation.

Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:

which is the same thing as

It follows from the zero-product property that either x = 2 {\displaystyle x=2} or x = − 5 {\displaystyle x=-5} are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,

has no real number solution since no real number squared equals −1.Sometimes a quadratic equation has a root of multiplicity 2, such as:

For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as

### Supplementary Text ###
Équations quatriques

Pour des discussions détaillées de certaines méthodes de résolution, voir :

Une équation quatrique a x⁴ + b x³ + c x² + d x + e = 0 avec a ≠ 0 peut être réduite à une équation quadratique par un changement de variable à condition qu'elle soit soit biquadratique (b = d = 0) soit quasi-palindromique (e = a, d = b).

Certaines équations cubiques et quatriques peuvent être résolues en utilisant la trigonométrie ou les fonctions hyperboliques.

### Draft ###

### 1. Query Parsing

"x² + 4x + 4 = 0" → standard form quadratic equation.

User seeks solutions. Direct computation needed.

French query. No translation ambiguity. Clear mathematical request.

### 2. Problem Classification

● Simple quadratic equation
● Complete square method applies directly  
● Single variable, real coefficients

Not complex. Basic algebra.

### 3. Solution Strategy

Standard approach: complete the square or quadratic formula.

For this specific case: x² + 4x + 4 = (x+2)² = 0

→ x = -2 with multiplicity 2.

Double root. Single solution.

### 4. Verification

Discriminant: Δ = b² - 4ac = 4² - 4×1×4 = 0

Zero discriminant → one real solution. Confirmed.

### 5. Answer Structure

State solution: x = -2 (multiplicity 2)
Explain method: completing square
Confirm discriminant: Δ = 0
Note: single real solution

### 6. Response Generation

French response appropriate.
Direct and complete.
No logical gaps.

∴ Solution: x = -2 with multiplicity 2.

Method: (x+2)² = 0 → x = -2.

Discriminant verification: Δ = 0.

Single real solution confirmed.

### Answer###
L'équation x² + 4x + 4 = 0 peut être résolue en utilisant la méthode de la complétion du carré.

Dans ce cas particulier, on peut factoriser l'équation sous la forme :

x² + 4x + 4 = (x + 2)² = 0

Cette factorisation montre que l'équation a une seule solution qui est x = -2, et cette solution apparaît deux fois (multiplicité 2).

Alternativement, cette équation peut être résolue en utilisant la formule quadratique, qui donne deux solutions :

Dans ce cas spécifique, la formule quadratique nous donne une seule valeur réelle pour x.

L'équation x² + 4x + 4 = 0 correspond donc à une équation quadratique qui a une seule solution réelle double : x = -2. Cette situation se produit lorsque le discriminant de l'équation quadratique est nul.