Resoudre Une Inéquation, Fonction Carrée

[myra]

Bonjour, je suis en seconde et j'arrive pas à faire cet exercice, je ne comprends vraiment pas. Merci de bien vouloir m'expliquer et m'aider. Je vous en remercie.

a) Résoudre l'inéquation x²>=2, en s'aidant de la courbe de la fonction carré dans un repère.

b)x désigne un nombre réel quelconque de l'intervalle [-1;3[ A quel intervalle appartient x²?

[pzorba75]

Bonjour, je suis en seconde et j'arrive pas à faire cet exercice, je ne comprends vraiment pas. Merci de bien vouloir m'expliquer et m'aider. Je vous en remercie.

a) Résoudre l'inéquation x²>=2, en s'aidant de la courbe de la fonction carré dans un repère.

x^2>=2

x^2-2>=0

(x+sqrt{2})(x-sqrt{2})>=0

solution x<=-sqrt{2} ou x>=+sqrt{2}

b)x désigne un nombre réel quelconque de l'intervalle [-1;3[ A quel intervalle appartient x²?

-1<= x 0

0<= x^2 <=1

et

0<= x <3

0<=x^2<3

soit en composant, il vient : 0<=x^2<3

[myra]

Merci beaucoup, mais "sqrt{2}" c'est quoi?

[myra]

comment dois-je comprendre ça?

[pzorba75]

sqrt est l'abréviation de la fonction racine carré (square root en anglais)

Voici ce qu'on trouve sur wikipedia.fr :

In mathematics, a square root of a number x is a number r such that r² = x, or, in other words, a number r whose square (the result of multiplying the number by itself, or r × r) is x.^[1] For example, 4 is a square root of 16 because 4×4=16.

Every non-negative real number x has a unique non-negative square root, called the principal square root, denoted by a radical sign as . For positive x, the principal square root can also be written in exponent notation, as x^1/2. For example, the principal square root of 9 is 3, denoted , because 3² = 3 × 3 = 9 and 3 is non-negative. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.

Every positive number x has two square roots. One of them is , which is positive, and the other , which is negative. Together, these two roots are denoted (see ± shorthand). Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where n and m are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.^[2] The particular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1.

The term whose root is being considered is known as the radicand. For example, in the expression , ab + 2 is the radicand. The radicand is the number or expression underneath the radical sign.

Cette abréviation est utilisée dans les traitements de texte pour "dessiner" l'expression racine carrée et dans les bons langages de programmation scientifique.

Pour faire simple, quand on tape sur un clavier, sans outil spécialisé d'importation d'image, on peut écrire :

pour racine carrée de x : x^1/2, ou sqrt(x)

pour racine cubique de x : x^1/3

pour racine n-ième de x : x^1/n

Bonne lecture.