{"id":665,"date":"2026-10-04T07:17:45","date_gmt":"2026-10-04T06:17:45","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/matematica\/equacao-quarto-grau\/"},"modified":"2026-10-04T07:17:45","modified_gmt":"2026-10-04T06:17:45","slug":"equacao-quarto-grau","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/equacao-quarto-grau\/","title":{"rendered":"Calculadora de Equa\u00e7\u00e3o do Quarto Grau"},"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\">Resolver Equa\u00e7\u00f5es do Quarto Grau<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"665\" data-favorite-title=\"Equa\u00e7\u00e3o do Quarto Grau\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/equacao-quarto-grau\/\" data-favorite-excerpt=\"Introduza os coeficientes a, b, c, d e e para resolver qualquer equa\u00e7\u00e3o do quarto grau ax\u2074 + bx\u00b3 + cx\u00b2 + dx + e = 0. A calculadora encontra as quatro ra\u00edzes, reais ou complexas, usando o m\u00e9todo de Ferrari.\" aria-label=\"Adicionar aos favoritos\" 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\/pt\/matematica\/equacao-quarto-grau\/\" method=\"POST\" class=\"calculator\" id=\"calculator-665\" data-post=\"665\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Campos de entrada<\/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=\"Recalcular\" id=\"calculate-button\" data-post=\"665\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/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> Seus resultados<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Raiz x\u2081<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>Raiz x\u2082<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" 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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\">Copiar link para este c\u00e1lculo<\/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>Deixar um coment\u00e1rio<\/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=\"665\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/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\">Curti <\/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\">\u00datil <\/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\">Confuso <\/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\">Decepcionado <\/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\">Impreciso <\/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>F\u00f3rmula da Equa\u00e7\u00e3o do Quarto Grau<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/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>Onde:<\/strong><\/p>\n<ul>\n<li>\\(a, b, c, d, e\\) \u2014 coeficientes reais do polin\u00f4mio<\/li>\n<li>\\(a\\) \u2014 coeficiente l\u00edder (n\u00e3o pode ser zero, caso contr\u00e1rio a equa\u00e7\u00e3o n\u00e3o \u00e9 do quarto grau)<\/li>\n<li>\\(x\\) \u2014 a inc\u00f3gnita a resolver<\/li>\n<\/ul>\n<p>Esta \u00e9 a forma geral de uma equa\u00e7\u00e3o polinomial de quarto grau. A sua resolu\u00e7\u00e3o anal\u00edtica usa o m\u00e9todo de Ferrari: primeiro a equa\u00e7\u00e3o \u00e9 reduzida a uma equa\u00e7\u00e3o do quarto grau reduzida (sem o termo \\(x^3\\)), depois resolve-se uma c\u00fabica resolutiva e as quatro ra\u00edzes s\u00e3o obtidas a partir de ra\u00edzes quadradas de express\u00f5es envolvendo a solu\u00e7\u00e3o dessa c\u00fabica. Esta calculadora executa todo o procedimento automaticamente.<\/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>Equa\u00e7\u00e3o do Quarto Grau \u2013 Exemplo de C\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> resolver \\(x^4 - 3x^3 + 3x^2 - x = 0\\), que corresponde aos valores predefinidos \\(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>Resultado:<\/strong> \\(x_1 = 0\\) e \\(x_2 = x_3 = x_4 = 1\\). A equa\u00e7\u00e3o tem uma raiz simples em 0 e uma raiz tripla em 1 \u2014 os mesmos quatro valores que a calculadora devolve.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como Utilizar a Calculadora<\/h2>\n<ol>\n<li>Introduza o coeficiente <strong>a<\/strong> do termo em \\(x^4\\). Tem de ser diferente de zero.<\/li>\n<li>Introduza o coeficiente <strong>b<\/strong> do termo em \\(x^3\\).<\/li>\n<li>Introduza o coeficiente <strong>c<\/strong> do termo em \\(x^2\\).<\/li>\n<li>Introduza o coeficiente <strong>d<\/strong> do termo em \\(x\\).<\/li>\n<li>Introduza o termo independente <strong>e<\/strong>. Use 0 se a equa\u00e7\u00e3o n\u00e3o tiver termo constante.<\/li>\n<li>Leia as quatro ra\u00edzes \\(x_1\\) a \\(x_4\\). As ra\u00edzes complexas aparecem na forma \\(p + qi\\) ou \\(p - qi\\).<\/li>\n<\/ol>\n<p>Use valores negativos para coeficientes com sinal de menos \u2014 por exemplo, para \\(x^4 - 3x^3\\) introduza a = 1 e b = \u22123. A equa\u00e7\u00e3o \u00e9 normalizada internamente, portanto qualquer coeficiente l\u00edder n\u00e3o nulo funciona, incluindo fra\u00e7\u00f5es como 0,5 (introduzida como 0.5 ou com v\u00edrgula decimal).<\/p>\n<h2>O Que \u00c9 uma Equa\u00e7\u00e3o do Quarto Grau?<\/h2>\n<p>Uma equa\u00e7\u00e3o do quarto grau \u00e9 uma equa\u00e7\u00e3o polinomial de grau quatro. Pelo teorema fundamental da \u00e1lgebra, ela tem sempre exatamente quatro ra\u00edzes no plano complexo, contadas com multiplicidade. Dependendo dos coeficientes, essas ra\u00edzes podem ser todas reais, todas complexas ou uma mistura das duas. Como os coeficientes introduzidos aqui s\u00e3o reais, quaisquer ra\u00edzes complexas aparecem sempre em <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/conjugado-numero-complexo\/\">pares conjugados<\/a> \\(p \\pm qi\\).<\/p>\n<p>O quarto grau \u00e9 o grau mais elevado para o qual existe uma solu\u00e7\u00e3o geral em radicais. O m\u00e9todo de Ferrari, descoberto no s\u00e9culo XVI, reduz o problema a uma <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/equacao-cubica\/\">equa\u00e7\u00e3o c\u00fabica<\/a> (a resolutiva) e depois termina com ra\u00edzes quadradas. \u00c9 por isso que a calculadora pode devolver valores alg\u00e9bricos exatos em vez de meras aproxima\u00e7\u00f5es num\u00e9ricas. A contrapartida \u00e9 que a express\u00e3o fechada completa \u00e9 extremamente longa, e \u00e9 por isso que ferramentas automatizadas como esta s\u00e3o usadas na pr\u00e1tica.<\/p>\n<p>Interpretar os resultados \u00e9 simples: as ra\u00edzes reais s\u00e3o os pontos onde a curva do polin\u00f4mio cruza o eixo das abscissas, enquanto as ra\u00edzes complexas indicam que a curva n\u00e3o toca o eixo nesses locais. Ra\u00edzes repetidas (como a raiz tripla no exemplo acima) significam que a curva \u00e9 tangente ao eixo nesse ponto. Se s\u00f3 precisa de uma ideia r\u00e1pida de quantas ra\u00edzes reais esperar antes de resolver, a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/regra-dos-sinais-de-descartes\/\">Regra dos Sinais de Descartes<\/a> \u00e9 uma verifica\u00e7\u00e3o \u00fatil.<\/p>\n<p>Rela\u00e7\u00f5es \u00fateis entre as ra\u00edzes e os coeficientes (f\u00f3rmulas de Vi\u00e8te) permitem verificar as respostas:<\/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>Pressupostos, Limita\u00e7\u00f5es e Dicas<\/h2>\n<ul>\n<li><strong>O coeficiente a n\u00e3o pode ser zero.<\/strong> Se introduzir a = 0, a calculadora reporta um erro porque a equa\u00e7\u00e3o deixa de ser do quarto grau. Se \\(a = 0\\) mas \\(b \\neq 0\\), trata-se na verdade de uma c\u00fabica; resolva-a com a calculadora de <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/equacao-cubica\/\">Equa\u00e7\u00e3o do Terceiro Grau<\/a>. Se tanto a como b forem zero, use a calculadora de <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/equacao-quadratica\/\">Equa\u00e7\u00e3o do Segundo Grau<\/a>.<\/li>\n<li><strong>As ra\u00edzes s\u00e3o arredondadas<\/strong> a tr\u00eas casas decimais, pelo que valores exatos como \\(\\sqrt{2}\\) aparecem como 1,414. Ra\u00edzes quase id\u00eanticas podem parecer iguais ap\u00f3s o arredondamento, mesmo diferindo ligeiramente.<\/li>\n<li><strong>Precis\u00e3o num\u00e9rica:<\/strong> para coeficientes extremamente grandes ou muito pequenos, a c\u00fabica resolutiva interm\u00e9dia pode perder precis\u00e3o. Reescalar a equa\u00e7\u00e3o (dividir todos os coeficientes por uma pot\u00eancia de 10) melhora a estabilidade.<\/li>\n<li><strong>Resultados complexos s\u00e3o normais:<\/strong> se a calculadora devolver pares como \\(2 + 3i\\) e \\(2 - 3i\\), o polin\u00f4mio n\u00e3o tem ra\u00edzes reais nessas posi\u00e7\u00f5es. Saiba mais sobre esses valores com a calculadora de <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/raiz-de-numero-complexo\/\">Raiz Complexa<\/a>.<\/li>\n<li><strong>Multiplicidades contam:<\/strong> s\u00e3o sempre apresentadas quatro ra\u00edzes, pelo que uma raiz dupla aparece duas vezes. No exemplo acima, \\(x = 1\\) \u00e9 listado tr\u00eas vezes porque \u00e9 uma raiz tripla.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introduza os coeficientes a, b, c, d e e para resolver qualquer equa\u00e7\u00e3o do quarto grau ax\u2074 + bx\u00b3 + cx\u00b2 + dx + e = 0. A calculadora encontra as quatro ra\u00edzes, reais ou complexas, usando o m\u00e9todo de Ferrari.<\/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":[8],"tags":[16],"class_list":["post-665","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/665","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=665"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/665\/revisions"}],"predecessor-version":[{"id":2780,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/665\/revisions\/2780"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=665"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=665"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}