Calculateur d'équation du second degré
Résolvez toute équation du second degré ax² + bx + c = 0 avec racines réelles ou complexes, et obtenez le discriminant, le sommet, l'axe de symétrie et la forme factorisée.
Comment utiliser cette calculatrice
- 1Write your equation in the standard form ax² + bx + c = 0, moving every term to one side.
- 2Enter a, b, and c — including their signs. A missing x term means b = 0.
- 3Read the roots, and check the discriminant to understand why you got real or complex answers.
Comment ça marche
La formule quadratique
x = (−b ± √(b² − 4ac)) ÷ 2a Discriminant : Δ = b² − 4ac Sommet : x = −b ÷ 2a, y = f(−b ÷ 2a) Somme des racines = −b ÷ a, Produit des racines = c ÷ a
La formule quadratique résout toute équation de la forme ax² + bx + c = 0. Le discriminant, la partie sous la racine (b² − 4ac), détermine le type de solution : positif, deux racines réelles distinctes ; nul, une racine double ; négatif, deux racines complexes conjuguées. Le sommet et l'axe de symétrie décrivent la parabole associée.
Exemple détaillé
Pour x² − 3x + 2 = 0 : le discriminant vaut (−3)² − 4·1·2 = 1, positif, il y a donc deux racines réelles. La formule donne x = (3 ± 1) ÷ 2, soit x = 2 et x = 1, et la forme factorisée est (x − 1)(x − 2).
Calculateur d'équation du second degré : le guide complet
What the discriminant tells you before you solve
The discriminant b² − 4ac is worth computing first, because it answers the shape of the problem in one number. Positive means the parabola crosses the x-axis twice and there are two distinct real solutions. Zero means it touches the axis at exactly one point — a repeated root, and the vertex sits on the axis. Negative means the parabola never reaches the axis, so there are no real solutions and the roots are a complex conjugate pair.
A discriminant that is a perfect square carries extra information: it means the quadratic factors cleanly over the rationals, so factoring by inspection will work and you did not need the formula at all.
Three ways to solve, and when each is fastest
Factoring is fastest when it works — if you can spot two numbers that multiply to ac and add to b, you are done in seconds. It only works when the roots are rational.
Completing the square is slower but it is where the quadratic formula comes from, and it is the method that converts a quadratic into vertex form a(x − h)² + k, which is what you want for graphing or optimisation.
The quadratic formula always works. It handles irrational roots, complex roots, and ugly decimal coefficients identically, which is why it is the method to reach for when the numbers are not friendly.
Reading the vertex and the parabola
The vertex is the turning point. Its x-coordinate is always −b ÷ 2a, sitting exactly halfway between the two roots because a parabola is symmetric about that line. Substituting that x back into the equation gives the y-coordinate.
The sign of a decides which way the curve opens: positive opens upward so the vertex is the minimum, negative opens downward so the vertex is the maximum. This is the whole basis of quadratic optimisation problems — maximum profit, maximum projectile height, minimum material cost all reduce to finding a vertex.
Vieta's formulas as a check
For any quadratic, the two roots must sum to −b ÷ a and multiply to c ÷ a. These are Vieta's formulas, and they give a free five-second check on any answer.
For x² − 5x + 6 = 0, the roots 3 and 2 sum to 5 (which is −(−5) ÷ 1) and multiply to 6 (which is 6 ÷ 1). If your roots fail either check, you have made an arithmetic slip. The check also works for complex roots, where the imaginary parts cancel in the sum and combine in the product.
Questions fréquentes
Que signifie le discriminant ?
Le signe de b² − 4ac indique le type de racines : positif, deux racines réelles distinctes ; nul, une racine réelle double ; négatif, deux racines complexes conjuguées sans intersection avec l'axe des x. Sa valeur mesure aussi l'écart entre les racines.
Que se passe-t-il si a vaut zéro ?
L'équation n'est alors plus du second degré mais linéaire (bx + c = 0) et n'a qu'une solution, x = −c ÷ b. La formule quadratique exige que a soit non nul, sinon il y aurait une division par zéro.
Une équation du second degré peut-elle n'avoir aucune solution réelle ?
Oui. Quand le discriminant est négatif, la parabole ne coupe pas l'axe des x et il n'y a pas de racine réelle ; les deux solutions sont des nombres complexes conjugués. Le calculateur les affiche sous la forme a ± bi.