{"id":61935,"date":"2021-05-31T08:38:13","date_gmt":"2021-05-31T06:38:13","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61935"},"modified":"2025-05-09T07:36:26","modified_gmt":"2025-05-09T04:36:26","slug":"polynomial-equation","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/polynomial-equation\/","title":{"rendered":"Cubic Equation Solver"},"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 class=\"text-lg font-display font-bold\">Cubic Equation<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"61935\" data-favorite-title=\"Cubic Equation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/polynomial-equation\/\" data-favorite-excerpt=\"This cubic equation solver helps you find the roots of any third-degree polynomial of the form $$ax\u00b3 + bx\u00b2 + cx + d = 0$$. Whether the equation has real or complex roots, the tool uses Cardano\u2019s method to calculate accurate solutions instantly. It\u2019s perfect for students, teachers, and professionals who need to solve cubic equations quickly and reliably.\r\n\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/polynomial-equation\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61935\" data-post=\"61935\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Coefficient a<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coefficient_a\" id=\"field-1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Coefficient b<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coefficient_b\" id=\"field-2\" step=\"any\" value=\"8\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Coefficient c<\/label><div class=\"term\">c<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coefficient_c\" id=\"field-3\" step=\"any\" value=\"16\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Coefficient d<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coefficient_d\" id=\"field-4\" step=\"any\" value=\"0\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61935\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/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> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Root x\u2081<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Root x\u2082<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Root x\u2083<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/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>Leave a Comment<\/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=\"61935\" aria-live=\"polite\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button 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\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button 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\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button 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\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button 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\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button 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\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/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><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<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>Binomial Multiplication Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\\[\\text{Given a cubic equation: } ax^3 + bx^2 + cx + d = 0 \\] \\[\\text{First, reduce it to depressed cubic form: } t^3 + pt + q = 0 \\]<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$t = x + \\frac{b}{3a}$$<\/li>\n<li>$$p = \\frac{3ac &#8211; b^2}{3a^2}$$<\/li>\n<li>$$q = \\frac{2b^3 &#8211; 9abc + 27a^2d}{27a^3}$$<\/li>\n<\/ul>\n<p>Then the root is: $$t = \\sqrt[3]{-\\frac{q}{2} + \\sqrt{\\left( \\frac{q}{2} \\right)^2 + \\left( \\frac{p}{3} \\right)^3}} + \\sqrt[3]{-\\frac{q}{2} &#8211; \\sqrt{\\left( \\frac{q}{2} \\right)^2 + \\left( \\frac{p}{3} \\right)^3}}$$<\/p>\n<p>Cardano\u2019s formula provides the real solution(s) of a depressed cubic. The full equation\u2019s roots are then found by back-substitution.<br \/>\r\n<\/section>\n<p>Cubic equations are fundamental in algebra and arise in various fields such as mechanics, electronics, and 3D modeling. This calculator uses a step-by-step method based on Cardano\u2019s formula to solve equations like $$2x\u00b3 &#8211; 4x\u00b2 &#8211; 22x + 24 = 0$$. After reducing the original cubic to a simpler \u201cdepressed\u201d form, the formula determines whether the roots are real or complex, single or multiple. The tool provides an instant and reliable way to analyze cubic behavior without manual calculations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This cubic equation solver helps you find the roots of any third-degree polynomial of the form $$ax\u00b3 + bx\u00b2 + cx + d = 0$$. Whether the equation has real or complex roots, the tool uses Cardano\u2019s method to calculate accurate solutions instantly. It\u2019s perfect for students, teachers, and professionals who need to solve cubic equations quickly and reliably.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-61935","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61935","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61935"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61935\/revisions"}],"predecessor-version":[{"id":63823,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61935\/revisions\/63823"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}