{"id":61479,"date":"2021-02-02T16:24:49","date_gmt":"2021-02-02T14:24:49","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61479"},"modified":"2026-09-30T20:47:02","modified_gmt":"2026-09-30T17:47:02","slug":"quartic-equation","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/quartic-equation\/","title":{"rendered":"Quartic 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 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Solve Quartic Equations<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61479\" data-favorite-title=\"Quartic Equation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/quartic-equation\/\" data-favorite-excerpt=\"Enter the coefficients a, b, c, d and e to solve any quartic equation ax\u2074 + bx\u00b3 + cx\u00b2 + dx + e = 0. The calculator finds all four roots, real or complex, using Ferrari&#039;s method.\" aria-label=\"Add to Favorites\" 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\/en\/mathematics\/quartic-equation\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61479\" data-post=\"61479\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">a<\/label><input type=\"number\" name=\"a\" id=\"field-1\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">b<\/label><input type=\"number\" name=\"b\" id=\"field-2\" placeholder=\"-3\" step=\"1\" value=\"-3\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">c<\/label><input type=\"number\" name=\"c\" id=\"field-3\" placeholder=\"3\" step=\"1\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">d<\/label><input type=\"number\" name=\"d\" id=\"field-4\" placeholder=\"-1\" step=\"1\" value=\"-1\"\/><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">e<\/label><input type=\"number\" name=\"e\" id=\"field-5\" placeholder=\"0\" step=\"1\" value=\"0\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61479\">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 type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/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 type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/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 type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Root x\u2084<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><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\">Copy link to this calculation<\/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>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=\"61479\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/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\">Like <\/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\">Helpful <\/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\">Confused <\/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\">Disappointed <\/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\">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><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>Quartic Equation 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\">\n\\[ ax^4 + bx^3 + cx^2 + dx + e = 0, \\quad a \\neq 0 \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a, b, c, d, e\\) \u2014 real coefficients of the polynomial<\/li>\n<li>\\(a\\) \u2014 leading coefficient (must not be zero, otherwise the equation is not quartic)<\/li>\n<li>\\(x\\) \u2014 the unknown to solve for<\/li>\n<\/ul>\n<p>This is the general form of a fourth-degree polynomial equation. Solving it analytically uses Ferrari's method: the equation is first reduced to a depressed quartic (no \\(x^3\\) term), then a resolvent cubic is solved, and the four roots are recovered from square roots of expressions involving that cubic's solution. This calculator carries out the full procedure automatically.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.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>Quartic Equation \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> solve \\(x^4 - 3x^3 + 3x^2 - x = 0\\), which corresponds to the default values \\(a = 1\\), \\(b = -3\\), \\(c = 3\\), \\(d = -1\\), \\(e = 0\\).<\/p>\n<p>\\[ x^4 - 3x^3 + 3x^2 - x = x(x^3 - 3x^2 + 3x - 1) = x(x - 1)^3 = 0 \\]<\/p>\n<p><strong>Result:<\/strong> \\(x_1 = 0\\) and \\(x_2 = x_3 = x_4 = 1\\). The equation has a simple root at 0 and a triple root at 1 \u2014 the same four values the calculator returns.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the coefficient <strong>a<\/strong> of the \\(x^4\\) term. It must be non-zero.<\/li>\n<li>Enter the coefficient <strong>b<\/strong> of the \\(x^3\\) term.<\/li>\n<li>Enter the coefficient <strong>c<\/strong> of the \\(x^2\\) term.<\/li>\n<li>Enter the coefficient <strong>d<\/strong> of the \\(x\\) term.<\/li>\n<li>Enter the constant term <strong>e<\/strong>. Use 0 if the equation has no constant.<\/li>\n<li>Read the four roots \\(x_1\\) through \\(x_4\\). Complex roots appear in the form \\(p + qi\\) or \\(p - qi\\).<\/li>\n<\/ol>\n<p>Use negative values for coefficients with minus signs \u2014 for example, for \\(x^4 - 3x^3\\) enter a = 1 and b = \u22123. The equation is normalized internally, so any non-zero leading coefficient works, including fractions such as 0.5 (entered as 0.5 or with decimal steps).<\/p>\n<h2>What Is a Quartic Equation?<\/h2>\n<p>A quartic equation is a polynomial equation of degree four. By the fundamental theorem of algebra it always has exactly four roots in the complex plane, counted with multiplicity. Depending on the coefficients, those roots may be all real, all complex, or a mix of the two. Because the coefficients entered here are real, any complex roots always come in <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\">conjugate pairs<\/a> \\(p \\pm qi\\).<\/p>\n<p>The quartic is the highest degree for which a general solution in radicals exists. Ferrari's method, discovered in the 16th century, reduces the problem to a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/cubic-equation\/\">cubic equation<\/a> (the resolvent) and then finishes with square roots. This is why the calculator can return exact algebraic values rather than mere numerical guesses. The trade-off is that the full closed-form expression is extremely long, which is why automated tools like this one are used in practice.<\/p>\n<p>Interpreting the results is straightforward: real roots are the points where the polynomial curve crosses the x-axis, while complex roots indicate the curve does not touch the axis at those locations. Repeated roots (like the triple root in the example above) mean the curve is tangent to the axis at that point. If you only need a quick sense of how many real roots to expect before solving, <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/descartes-rule-of-signs\/\">Descartes' Rule of Signs<\/a> is a handy check.<\/p>\n<p>Useful relationships between the roots and the coefficients (Vieta's formulas) let you verify the answers:<\/p>\n<ul>\n<li>\\(x_1 + x_2 + x_3 + x_4 = -b\/a\\)<\/li>\n<li>\\(x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4 = -d\/a\\)<\/li>\n<li>\\(x_1 x_2 x_3 x_4 = e\/a\\)<\/li>\n<\/ul>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Coefficient a must not be zero.<\/strong> If you enter a = 0, the calculator reports an error because the equation is no longer quartic. If \\(a = 0\\) but \\(b \\neq 0\\), it is actually a cubic; solve it with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/polynomial-equation\/\">Cubic Equation calculator<\/a> instead. If both a and b are zero, use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/\">Quadratic Equation calculator<\/a>.<\/li>\n<li><strong>Roots are rounded<\/strong> to three decimal places, so exact values like \\(\\sqrt{2}\\) appear as 1.414. Near-identical roots may look the same after rounding even when they differ slightly.<\/li>\n<li><strong>Numerical precision:<\/strong> for extremely large or tiny coefficients the intermediate resolvent cubic can lose precision. Rescaling the equation (dividing all coefficients by a power of 10) improves stability.<\/li>\n<li><strong>Complex output<\/strong> is normal: if the calculator returns pairs such as \\(2 + 3i\\) and \\(2 - 3i\\), the polynomial has no real roots at those positions. Read more about such values with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/\">Complex Root<\/a> calculator.<\/li>\n<li><strong>Multiplicity counts:<\/strong> four roots are always reported, so a double root appears twice. In the example above, \\(x = 1\\) is listed three times because it is a triple root.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter the coefficients a, b, c, d and e to solve any quartic equation ax\u2074 + bx\u00b3 + cx\u00b2 + dx + e = 0. The calculator finds all four roots, real or complex, using Ferrari's method.<\/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-61479","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\/61479","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=61479"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61479\/revisions"}],"predecessor-version":[{"id":66282,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61479\/revisions\/66282"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}