{"id":211,"date":"2026-10-03T06:04:09","date_gmt":"2026-10-03T03:04:09","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=211"},"modified":"2026-10-03T06:04:09","modified_gmt":"2026-10-03T03:04:09","slug":"discriminant-equation-second-degre","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/discriminant-equation-second-degre\/","title":{"rendered":"Calculateur du 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\">Calculez le discriminant de votre \u00e9quation<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"211\" data-favorite-title=\"Discriminant d\u2019une \u00e9quation du second degr\u00e9\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/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 son discriminant \u0394 = b\u00b2 \u2212 4ac et voir imm\u00e9diatement si elle poss\u00e8de deux racines r\u00e9elles, une racine double ou deux racines 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\/discriminant-equation-second-degre\/\" method=\"POST\" class=\"calculator\" id=\"calculator-211\" data-post=\"211\"><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=\"211\">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=\"211\" 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\\), \\(b\\), \\(c\\) \u2014 les coefficients de l\u2019\u00e9quation du second degr\u00e9 \\(ax^2 + bx + c = 0\\), avec \\(a \\neq 0\\)<\/li>\n<li>\\(\\Delta\\) (delta) \u2014 le discriminant<\/li>\n<\/ul>\n<p>Le discriminant est l\u2019expression sous la racine carr\u00e9e dans la formule de r\u00e9solution \\(x = \\dfrac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\\). Son signe d\u00e9termine la nature et le nombre de racines de l\u2019\u00e9quation&nbsp;:<\/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 l\u2019\u00e9quation \\(x^2 + 2x + 1 = 0\\), o\u00f9 \\(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&nbsp;:<\/strong> \\(\\Delta = 0\\). L\u2019\u00e9quation poss\u00e8de une racine r\u00e9elle double, \\(x = -1\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur de discriminant<\/h2>\n<ol>\n<li>Saisissez le coefficient \\(a\\) de \\(x^2\\) dans le champ <strong>a (coefficient de x\u00b2)<\/strong>. Cette valeur ne doit pas \u00eatre nulle.<\/li>\n<li>Saisissez le coefficient \\(b\\) de \\(x\\) dans le champ <strong>b (coefficient de x)<\/strong>. Utilisez un signe n\u00e9gatif si le terme est soustrait, par exemple \\(-3\\) pour \\(-3x\\).<\/li>\n<li>Saisissez le terme constant \\(c\\) dans le champ <strong>c (constante)<\/strong>, avec son signe.<\/li>\n<li>Consultez le r\u00e9sultat&nbsp;: le calculateur affiche le discriminant \\(\\Delta\\) arrondi \u00e0 trois d\u00e9cimales.<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que le discriminant d\u2019une \u00e9quation du second degr\u00e9&nbsp;?<\/h2>\n<p>Toute \u00e9quation du second degr\u00e9 peut s\u2019\u00e9crire sous la forme canonique \\(ax^2 + bx + c = 0\\), avec \\(a \\neq 0\\). Le discriminant \\(\\Delta = b^2 - 4ac\\) est un seul nombre qui indique combien de solutions poss\u00e8de l\u2019\u00e9quation et quelle est leur nature, sans avoir \u00e0 r\u00e9soudre l\u2019\u00e9quation.<\/p>\n<p>Cela tient \u00e0 la formule de r\u00e9solution \\(x = \\dfrac{-b \\pm \\sqrt{\\Delta}}{2a}\\). La seule partie dont la nature peut changer est \\(\\sqrt{\\Delta}\\). Si \\(\\Delta\\) est positif, sa racine carr\u00e9e est un nombre r\u00e9el et le signe \\(\\pm\\) donne deux racines r\u00e9elles distinctes. Si \\(\\Delta = 0\\), la racine carr\u00e9e s\u2019annule et les deux racines se confondent en une seule valeur, \\(x = -\\dfrac{b}{2a}\\). Si \\(\\Delta\\) est n\u00e9gatif, la racine carr\u00e9e d\u2019un nombre n\u00e9gatif est imaginaire&nbsp;: les deux racines forment alors une paire de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-conjugue-complexe\/\">nombres complexes conjugu\u00e9s<\/a>. Vous pouvez \u00e9tudier ces racines avec le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/racines-nombre-complexe\/\">calculateur de racines complexes<\/a>.<\/p>\n<p>G\u00e9om\u00e9triquement, le discriminant indique combien de fois la parabole \\(y = ax^2 + bx + c\\) coupe l\u2019axe des x : \\(\\Delta &gt; 0\\) correspond \u00e0 deux points d\u2019intersection, \\(\\Delta = 0\\) signifie que la parabole touche l\u2019axe en son sommet, et \\(\\Delta &lt; 0\\) signifie qu\u2019elle ne rencontre pas l\u2019axe.<\/p>\n<p>V\u00e9rifier d\u2019abord le discriminant est une pratique courante en alg\u00e8bre, en physique et en ing\u00e9nierie. Il indique imm\u00e9diatement si un probl\u00e8me poss\u00e8de une solution unique (\\(\\Delta = 0\\)), deux solutions possibles (\\(\\Delta &gt; 0\\)) ou aucune solution r\u00e9elle (\\(\\Delta &lt; 0\\)), avant de poursuivre les calculs. Si vous avez ensuite besoin des racines elles-m\u00eames, le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/equation-second-degre-2\/\">calculateur d\u2019\u00e9quations du second degr\u00e9<\/a> r\u00e9sout enti\u00e8rement l\u2019\u00e9quation.<\/p>\n<h2>Interpr\u00e9ter le r\u00e9sultat&nbsp;: tableau r\u00e9capitulatif<\/h2>\n<table>\n<thead>\n<tr>\n<th>Discriminant<\/th>\n<th>Nombre de racines r\u00e9elles<\/th>\n<th>Nature des racines<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\\(\\Delta &gt; 0\\)<\/td>\n<td>2 distinctes<\/td>\n<td>Rationnelles si \\(\\Delta\\) est un carr\u00e9 parfait, irrationnelles sinon<\/td>\n<\/tr>\n<tr>\n<td>\\(\\Delta = 0\\)<\/td>\n<td>1 (double)<\/td>\n<td>Rationnelle : \\(x = -\\frac{b}{2a}\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(\\Delta &lt; 0\\)<\/td>\n<td>0<\/td>\n<td>Deux racines complexes conjugu\u00e9es<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Conditions, limites et conseils<\/h2>\n<ul>\n<li><strong>Le coefficient \\(a\\) doit \u00eatre non nul.<\/strong> Si \\(a = 0\\), l\u2019\u00e9quation est du premier degr\u00e9 (\\(bx + c = 0\\)), et non du second degr\u00e9&nbsp;: la formule du discriminant ne s\u2019applique pas.<\/li>\n<li><strong>Tenez compte du signe des coefficients.<\/strong> Pour \\(x^2 - 5x + 6 = 0\\), saisissez \\(a = 1\\), \\(b = -5\\), \\(c = 6\\). Saisir \\(b = 5\\) donne un discriminant incorrect (\\(49\\) au lieu de \\(1\\)).<\/li>\n<li><strong>Les fractions et les nombres d\u00e9cimaux conviennent.<\/strong> Les champs acceptent les valeurs d\u00e9cimales : vous pouvez donc utiliser des coefficients comme \\(a = 0.5\\) ou \\(b = -1.25\\).<\/li>\n<li><strong>Arrondi.<\/strong> Le r\u00e9sultat est arrondi \u00e0 trois d\u00e9cimales&nbsp;; une valeur affich\u00e9e comme \\(0\\) peut en r\u00e9alit\u00e9 \u00eatre un tr\u00e8s petit nombre non nul si vous avez saisi des coefficients arrondis.<\/li>\n<li><strong>Le discriminant classe les racines, mais ne les calcule pas.<\/strong> Pour trouver les racines elles-m\u00eames, utilisez le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/equation-second-degre-2\/\">calculateur d\u2019\u00e9quations du second degr\u00e9<\/a>. Pour les polyn\u00f4mes de degr\u00e9 sup\u00e9rieur, consultez les calculateurs d\u2019<a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/resolution-equations-cubiques\/\">\u00e9quations du troisi\u00e8me degr\u00e9<\/a> et de la <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/regle-des-signes-de-descartes\/\">r\u00e8gle des signes de Descartes<\/a>, qui analysent le nombre de racines des \u00e9quations du troisi\u00e8me degr\u00e9 et des \u00e9quations polynomiales en g\u00e9n\u00e9ral.<\/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 son discriminant \u0394 = b\u00b2 \u2212 4ac et voir imm\u00e9diatement si elle poss\u00e8de deux racines r\u00e9elles, une racine double ou deux racines 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-211","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\/211","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=211"}],"version-history":[{"count":6,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/211\/revisions"}],"predecessor-version":[{"id":5361,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/211\/revisions\/5361"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}