{"id":212,"date":"2026-10-03T06:04:09","date_gmt":"2026-10-03T03:04:09","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=212"},"modified":"2026-10-03T06:04:09","modified_gmt":"2026-10-03T03:04:09","slug":"calculateur-discriminant-equation-second-degre","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-discriminant-equation-second-degre\/","title":{"rendered":"Calculateur de discriminant d\u2019une \u00e9quation du second degr\u00e9"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Calculateur du discriminant d\u2019une \u00e9quation du second degr\u00e9<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"212\" data-favorite-title=\"Discriminant d\u2019une \u00e9quation du second degr\u00e9\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-discriminant-equation-second-degre\/\" data-favorite-excerpt=\"Saisissez les coefficients a, b et c d\u2019une \u00e9quation du second degr\u00e9 ax\u00b2 + bx + c = 0 pour calculer instantan\u00e9ment son discriminant \u0394 = b\u00b2 \u2212 4ac et savoir combien elle poss\u00e8de de racines r\u00e9elles ou complexes.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-discriminant-equation-second-degre\/\" method=\"POST\" class=\"calculator\" id=\"calculator-212\" data-post=\"212\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">a (coefficient de x\u00b2)<\/label><input type=\"number\" name=\"a_coefficient_de_x\" id=\"field-1\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">b (coefficient de x)<\/label><input type=\"number\" name=\"b_coefficient_de_x\" id=\"field-2\" placeholder=\"2\" step=\"1\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">c (terme constant)<\/label><input type=\"number\" name=\"c_terme_constant\" id=\"field-3\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"212\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Vos r\u00e9sultats<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Discriminant<\/span><span class=\"result-term\">\u0394<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copier le lien vers ce calcul<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Laisser un commentaire<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"212\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Que pensez-vous de ce r\u00e9sultat\u00a0?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">J\u2019aime <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Utile <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confus <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">D\u00e9\u00e7u <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inexact <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Formule du discriminant d\u2019une \u00e9quation du second degr\u00e9<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\Delta = b^2 - 4ac \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(a\\) \u2014 coefficient de \\(x^2\\) (ne doit pas \u00eatre nul)<\/li>\n<li>\\(b\\) \u2014 coefficient de \\(x\\)<\/li>\n<li>\\(c\\) \u2014 terme constant<\/li>\n<li>\\(\\Delta\\) (delta) \u2014 discriminant de l\u2019\u00e9quation \\(ax^2 + bx + c = 0\\)<\/li>\n<\/ul>\n<p>Le discriminant est l\u2019expression sous la racine carr\u00e9e dans la formule de r\u00e9solution d\u2019une \u00e9quation du second degr\u00e9. Son signe indique la nature des racines sans qu\u2019il soit n\u00e9cessaire de r\u00e9soudre l\u2019\u00e9quation :<\/p>\n<ul>\n<li>\\(\\Delta &gt; 0\\) \u2014 deux racines r\u00e9elles distinctes<\/li>\n<li>\\(\\Delta = 0\\) \u2014 une racine r\u00e9elle double<\/li>\n<li>\\(\\Delta &lt; 0\\) \u2014 deux racines complexes conjugu\u00e9es<\/li>\n<\/ul>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Discriminant \u2013 exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> calculons le discriminant de \\(x^2 + 2x + 1 = 0\\), avec \\(a = 1\\), \\(b = 2\\) et \\(c = 1\\).<\/p>\n<p>\\[ \\Delta = b^2 - 4ac = 2^2 - 4 \\cdot 1 \\cdot 1 = 4 - 4 = 0 \\]<\/p>\n<p><strong>R\u00e9sultat :<\/strong> \\(\\Delta = 0\\). L\u2019\u00e9quation poss\u00e8de exactement une racine r\u00e9elle double, \\(x = -1\\).<\/p>\n<p><strong>Autre exemple&nbsp;:<\/strong> pour \\(x^2 - 3x + 2 = 0\\), avec \\(a = 1\\), \\(b = -3\\) et \\(c = 2\\) :<\/p>\n<p>\\[ \\Delta = (-3)^2 - 4 \\cdot 1 \\cdot 2 = 9 - 8 = 1 \\]<\/p>\n<p><strong>R\u00e9sultat :<\/strong> \\(\\Delta = 1 &gt; 0\\), donc il y a deux racines r\u00e9elles distinctes.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Saisissez le coefficient de \\(x^2\\) dans le champ <strong>a (coefficient de x\u00b2)<\/strong>.<\/li>\n<li>Saisissez le coefficient de \\(x\\) dans le champ <strong>b (coefficient de x)<\/strong>.<\/li>\n<li>Saisissez le terme constant dans le champ <strong>c (constante)<\/strong>.<\/li>\n<li>Le r\u00e9sultat <strong>Discriminant<\/strong> se met \u00e0 jour automatiquement et affiche la valeur de \\(\\Delta\\).<\/li>\n<\/ol>\n<p>Vous pouvez utiliser tous les nombres r\u00e9els, y compris les nombres n\u00e9gatifs et d\u00e9cimaux. Assurez-vous que \\(a \\neq 0\\) ; si \\(a = 0\\), l\u2019\u00e9quation n\u2019est pas du second degr\u00e9, mais du premier degr\u00e9, et le discriminant n\u2019est pas d\u00e9fini.<\/p>\n<h2>Qu\u2019est-ce que le discriminant ?<\/h2>\n<p>Toute \u00e9quation du second degr\u00e9 peut s\u2019\u00e9crire sous la forme standard \\(ax^2 + bx + c = 0\\). Ses racines sont donn\u00e9es par la formule suivante :<\/p>\n<p>\\[ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\]<\/p>\n<p>L\u2019expression sous la racine carr\u00e9e, \\(\\Delta = b^2 - 4ac\\), est appel\u00e9e discriminant (du latin \\discernere\\, \u00ab&nbsp;distinguer&nbsp;\u00bb), car elle permet de distinguer les diff\u00e9rents types de racines possibles.<\/p>\n<p>Le principe est simple : la formule utilise la racine carr\u00e9e de \\(\\Delta\\). Si \\(\\Delta &gt; 0\\), cette racine carr\u00e9e est un nombre r\u00e9el positif : le signe \\(\\pm\\) donne donc deux racines r\u00e9elles diff\u00e9rentes. Si \\(\\Delta = 0\\), les deux possibilit\u00e9s de la formule se r\u00e9duisent \u00e0 la seule valeur \\(x = -b\/2a\\), ce qui donne une racine double. Si \\(\\Delta &lt; 0\\), la racine carr\u00e9e d\u2019un nombre n\u00e9gatif n\u2019est pas r\u00e9elle ; les deux racines forment alors une paire de nombres complexes conjugu\u00e9s, de m\u00eame partie r\u00e9elle \\(-b\/2a\\) et de parties imaginaires oppos\u00e9es.<\/p>\n<p>Le discriminant est un outil de diagnostic rapide : avant m\u00eame de r\u00e9soudre l\u2019\u00e9quation, une soustraction permet de savoir si la parabole \\(y = ax^2 + bx + c\\) coupe l\u2019axe des abscisses en deux points, le touche en un seul point ou ne l\u2019atteint jamais. On l\u2019utilise aussi bien en alg\u00e8bre scolaire qu\u2019en ing\u00e9nierie, en physique et en optimisation, o\u00f9 savoir si un syst\u00e8me admet des solutions r\u00e9elles compte autant que les solutions elles-m\u00eames.<\/p>\n<h2>Interpr\u00e9ter le r\u00e9sultat<\/h2>\n<ul>\n<li><strong>\\(\\Delta &gt; 0\\) :<\/strong> deux racines r\u00e9elles distinctes, \\(x = \\frac{-b \\pm \\sqrt{\\Delta}}{2a}\\). La parabole coupe l\u2019axe des abscisses en deux points.<\/li>\n<li><strong>\\(\\Delta = 0\\) :<\/strong> une racine r\u00e9elle de multiplicit\u00e9 deux, \\(x = -\\frac{b}{2a}\\). La parabole est tangente \u00e0 l\u2019axe des abscisses en son sommet.<\/li>\n<li><strong>\\(\\Delta &lt; 0\\) :<\/strong> aucune racine r\u00e9elle&nbsp;; deux racines complexes conjugu\u00e9es \\(x = \\frac{-b \\pm i\\sqrt{|\\Delta|}}{2a}\\). La parabole se situe enti\u00e8rement au-dessus ou au-dessous de l\u2019axe des abscisses. Vous pouvez explorer ces racines avec le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/racines-nombre-complexe\/\">calculateur de racines complexes<\/a> et le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculatrice-nombres-complexes\/\">calculateur de nombres complexes<\/a>.<\/li>\n<\/ul>\n<h2>Hypoth\u00e8ses, limites et conseils<\/h2>\n<ul>\n<li>Le calculateur suppose qu\u2019il s\u2019agit bien d\u2019une \u00e9quation du second degr\u00e9, donc que \\(a \\neq 0\\). Si \\(a = 0\\), l\u2019\u00e9quation se r\u00e9duit \u00e0 \\(bx + c = 0\\) et doit \u00eatre r\u00e9solue comme une \u00e9quation du premier degr\u00e9.<\/li>\n<li>Le r\u00e9sultat est arrondi \u00e0 trois d\u00e9cimales. Si les coefficients sont tr\u00e8s grands, la valeur exacte de \\(\\Delta\\) peut comporter plus de chiffres que ceux affich\u00e9s.<\/li>\n<li>C\u2019est le signe de \\(\\Delta\\), et non sa valeur absolue, qui d\u00e9termine le type de racines. Une valeur positive de \\(\\Delta\\) plus grande signifie simplement que les deux racines r\u00e9elles sont plus \u00e9loign\u00e9es l\u2019une de l\u2019autre.<\/li>\n<li>\\(\\Delta\\) indique <em>combien<\/em> il existe de racines de chaque type, mais pas leurs valeurs. Pour trouver les racines, appliquez la formule de r\u00e9solution ou utilisez le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/equation-second-degre-2\/\">calculateur d\u2019\u00e9quations du second degr\u00e9<\/a>.<\/li>\n<li>Pour les polyn\u00f4mes de degr\u00e9 sup\u00e9rieur, d\u2019autres outils sont n\u00e9cessaires : consultez le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/resolution-equations-cubiques\/\">calculateur d\u2019\u00e9quations du troisi\u00e8me degr\u00e9<\/a> pour les \u00e9quations de degr\u00e9 trois.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Saisissez les coefficients a, b et c d\u2019une \u00e9quation du second degr\u00e9 ax\u00b2 + bx + c = 0 pour calculer instantan\u00e9ment son discriminant \u0394 = b\u00b2 \u2212 4ac et savoir combien elle poss\u00e8de de racines r\u00e9elles ou complexes.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[52],"class_list":["post-212","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-algebre"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/212","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=212"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/212\/revisions"}],"predecessor-version":[{"id":5983,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/212\/revisions\/5983"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=212"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}