{"id":160,"date":"2026-10-03T06:04:43","date_gmt":"2026-10-03T03:04:43","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=160"},"modified":"2026-10-03T06:04:43","modified_gmt":"2026-10-03T03:04:43","slug":"calculateur-racine-cubique","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-racine-cubique\/","title":{"rendered":"Calculateur de racine cubique"},"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 d\u2019estimation de racine cubique<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"160\" data-favorite-title=\"Racine cubique\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-racine-cubique\/\" data-favorite-excerpt=\"Calculez la variable d\u00e9cal\u00e9e x\u2099 = y\u2099 \u2212 b\/(3a), obtenue par le changement de variable qui \u00e9limine le terme quadratique de l\u2019\u00e9quation cubique ax\u00b3 + bx\u00b2 + cx + d = 0. Saisissez y\u2099 ainsi que les coefficients b et a pour obtenir instantan\u00e9ment un r\u00e9sultat pr\u00e9cis.\" 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-racine-cubique\/\" method=\"POST\" class=\"calculator\" id=\"calculator-160\" data-post=\"160\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">y\u2099<\/label><input type=\"number\" name=\"y\" id=\"field-1\" placeholder=\"8\" step=\"0.01\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Constante r\u00e9elle b<\/label><input type=\"number\" name=\"constante_relle_b\" id=\"field-2\" placeholder=\"2\" step=\"0.01\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Constante r\u00e9elle a<\/label><input type=\"number\" name=\"constante_relle_a\" id=\"field-3\" placeholder=\"1\" step=\"0.01\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"160\">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>x\u2099<\/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=\"160\" 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\" 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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>Racine cubique<\/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\\[ x_n = y_n - \\frac{b}{3a} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(x_n\\) \u2014 la variable d\u00e9cal\u00e9e (la nouvelle valeur apr\u00e8s le changement de variable),<\/li>\n<li>\\(y_n\\) \u2014 la valeur actuelle ou l\u2019approximation initiale,<\/li>\n<li>\\(a\\) \u2014 le coefficient dominant de l\u2019\u00e9quation cubique (il doit \u00eatre non nul),<\/li>\n<li>\\(b\\) \u2014 le coefficient du terme quadratique.<\/li>\n<\/ul>\n<p>Il s\u2019agit du changement de variable classique utilis\u00e9 pour r\u00e9soudre une \u00e9quation cubique \\(ax^3 + bx^2 + cx + d = 0\\). Remplacer \\(x\\) par \\(y - \\frac{b}{3a}\\) \u00e9limine le terme en \\(y^2\\) et ram\u00e8ne l\u2019\u00e9quation \u00e0 une forme \u00ab&nbsp;d\u00e9prim\u00e9e&nbsp;\u00bb plus simple, \\(ay^3 + py + q = 0\\), qui est le point de d\u00e9part de la formule de Cardan.<\/p>\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>Racine cubique \u2013 Exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> vous simplifiez l\u2019\u00e9quation cubique \\(x^3 + 2x^2 - 5x + 1 = 0\\) et cherchez la valeur d\u00e9cal\u00e9e correspondant \u00e0 une estimation actuelle de \\(y_n = 8\\).<\/p>\n<p>\\[ x_n = y_n - \\frac{b}{3a} = 8 - \\frac{2}{3 \\cdot 1} = 8 - 0.6667 \\approx 7.333 \\]<\/p>\n<p><strong>R\u00e9sultat&nbsp;:<\/strong> \\(x_n \\approx 7.333\\). Le changement de variable d\u00e9cale \\(y_n\\) de \\(-\\frac{b}{3a}\\), soit exactement le d\u00e9calage n\u00e9cessaire pour \u00e9liminer le terme quadratique de l\u2019\u00e9quation.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Saisissez la valeur de \\(y_n\\) dans le premier champ&nbsp;: il s\u2019agit de votre valeur actuelle, de votre estimation initiale ou du nombre que vous souhaitez d\u00e9caler.<\/li>\n<li>Saisissez la <strong>constante r\u00e9elle b<\/strong>&nbsp;: le coefficient du terme quadratique de votre \u00e9quation cubique.<\/li>\n<li>Saisissez la <strong>constante r\u00e9elle a<\/strong>&nbsp;: le coefficient dominant. Il ne doit pas \u00eatre nul, car la formule comporte une division par \\(3a\\).<\/li>\n<li>Lisez le r\u00e9sultat \\(x_n\\), affich\u00e9 avec un arrondi \u00e0 trois d\u00e9cimales.<\/li>\n<\/ol>\n<h2>Que signifie ce calcul ?<\/h2>\n<p>Une \u00e9quation cubique sous sa forme g\u00e9n\u00e9rale \\(ax^3 + bx^2 + cx + d = 0\\) est difficile \u00e0 r\u00e9soudre directement. Il y a plusieurs si\u00e8cles, les math\u00e9maticiens ont d\u00e9couvert qu\u2019un simple changement de variable rendait le probl\u00e8me beaucoup plus facile \u00e0 traiter. En d\u00e9calant l\u2019inconnue \u00e0 l\u2019aide du changement de variable \\(x = y - \\frac{b}{3a}\\), le terme en \\(y^2\\) s\u2019annule. On obtient ainsi une <em>\u00e9quation cubique d\u00e9prim\u00e9e<\/em> \\(y^3 + py + q = 0\\) (apr\u00e8s division par \\(a\\)). Cette \u00e9quation simplifi\u00e9e peut ensuite \u00eatre r\u00e9solue \u00e0 l\u2019aide de la formule de Cardan ou de m\u00e9thodes num\u00e9riques.<\/p>\n<p>Le d\u00e9calage \\(-\\frac{b}{3a}\\) a \u00e9galement une signification g\u00e9om\u00e9trique&nbsp;: il d\u00e9place l\u2019origine au point d\u2019inflexion de la courbe cubique. Toute fonction cubique \\(f(x) = ax^3 + bx^2 + cx + d\\) pr\u00e9sente une sym\u00e9trie centrale autour de son point d\u2019inflexion, qui se trouve toujours en \\(x = -\\frac{b}{3a}\\). Apr\u00e8s le d\u00e9calage, la courbe a la forme d\u2019un terme cubique \u00ab pur \u00bb \\(y^3\\) auquel s\u2019ajoute un terme lin\u00e9aire : c\u2019est pourquoi le coefficient de \\(y^2\\) dispara\u00eet.<\/p>\n<p>Bien que cette page s\u2019intitule \u00ab&nbsp;Racine cubique&nbsp;\u00bb, le calcul pr\u00e9sent\u00e9 n\u2019est qu\u2019une \u00e9tape de la recherche des racines d\u2019un polyn\u00f4me du troisi\u00e8me degr\u00e9. Si vous cherchez la racine cubique d\u2019un nombre \\(n\\) (c\u2019est-\u00e0-dire \\(\\sqrt[3]{n}\\)), vous pouvez d\u00e9finir \\(a = 1\\), \\(b = 0\\) et \\(y_n = n\\) : le d\u00e9calage est alors nul et \\(x_n = y_n\\). Pour r\u00e9soudre des \u00e9quations cubiques \u00e9tape par \u00e9tape, utilisez notre <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/resolution-equations-cubiques\/\">calculateur d\u2019\u00e9quation cubique<\/a>. Pour les polyn\u00f4mes du quatri\u00e8me degr\u00e9, consultez le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/equation-quartique\/\">calculateur d\u2019\u00e9quation quartique<\/a>.<\/p>\n<h2>Hypoth\u00e8ses, limites et conseils<\/h2>\n<ul>\n<li><strong>\\(a\\) doit \u00eatre non nul.<\/strong> La formule comporte une division par \\(3a\\) ; si vous saisissez \\(a = 0\\), l\u2019expression n\u2019est pas d\u00e9finie.<\/li>\n<li><strong>Il s\u2019agit d\u2019une seule \u00e9tape, pas d\u2019une r\u00e9solution compl\u00e8te.<\/strong> Le r\u00e9sultat \\(x_n\\) est une valeur d\u00e9cal\u00e9e, pas une racine de l\u2019\u00e9quation \u00e0 lui seul. Pour achever la r\u00e9solution d\u2019une \u00e9quation cubique d\u00e9prim\u00e9e, utilisez la m\u00e9thode de Cardan ou une m\u00e9thode num\u00e9rique comme celle de Newton-Raphson.<\/li>\n<li><strong>Les signes comptent.<\/strong> Si \\(b\\) est n\u00e9gatif, le d\u00e9calage \\(-\\frac{b}{3a}\\) devient positif, si bien que \\(x_n &gt; y_n\\). V\u00e9rifiez les signes de vos coefficients avant de les saisir.<\/li>\n<li><strong>Arrondi.<\/strong> Le r\u00e9sultat est arrondi \u00e0 trois d\u00e9cimales. Cela suffit pour la plupart des exercices scolaires et des applications d\u2019ing\u00e9nierie. Si vous avez besoin de davantage de chiffres, conservez la fraction exacte \\(y_n - \\frac{b}{3a}\\) dans la suite de vos calculs.<\/li>\n<li><strong>Coefficients r\u00e9els uniquement.<\/strong> Le calculateur est con\u00e7u pour des valeurs r\u00e9elles de \\(y_n\\), \\(a\\) et \\(b\\). Si votre \u00e9quation cubique a des racines complexes, le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/racines-nombre-complexe\/\">calculateur de racines complexes<\/a> peut vous aider \u00e0 les \u00e9tudier.<\/li>\n<\/ul>\n<h2>Rep\u00e8res rapides&nbsp;: d\u00e9calages courants<\/h2>\n<table>\n<thead>\n<tr>\n<th>\\(y_n\\)<\/th>\n<th>\\(a\\)<\/th>\n<th>\\(b\\)<\/th>\n<th>\\(x_n = y_n - \\frac{b}{3a}\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>8<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>7,333<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>8,000<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>\\(-1\\)<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>2<\/td>\n<td>6<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>1<\/td>\n<td>\\(-3\\)<\/td>\n<td>11,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>La troisi\u00e8me ligne illustre un fait utile&nbsp;: si l\u2019on part de \\(y_n = 0\\), le r\u00e9sultat est tout simplement l\u2019abscisse du point d\u2019inflexion \\(-\\frac{b}{3a}\\) de la courbe cubique. Pour d\u2019autres outils sur des sujets connexes, consultez le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/racine-quatrieme\/\">calculateur de racine quatri\u00e8me<\/a> ou le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/equation-second-degre-2\/\">calculateur d\u2019\u00e9quation du second degr\u00e9<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculez la variable d\u00e9cal\u00e9e x\u2099 = y\u2099 \u2212 b\/(3a), obtenue par le changement de variable qui \u00e9limine le terme quadratique de l\u2019\u00e9quation cubique ax\u00b3 + bx\u00b2 + cx + d = 0. Saisissez y\u2099 ainsi que les coefficients b et a pour obtenir instantan\u00e9ment un r\u00e9sultat pr\u00e9cis.<\/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-160","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\/160","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=160"}],"version-history":[{"count":6,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":5188,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/160\/revisions\/5188"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}