{"id":1678234,"date":"2025-12-22T11:17:00","date_gmt":"2025-12-22T14:17:00","guid":{"rendered":"https:\/\/www.estrategiaconcursos.com.br\/blog\/?p=1678234"},"modified":"2025-12-12T15:43:06","modified_gmt":"2025-12-12T18:43:06","slug":"inequacoes-2-grau","status":"publish","type":"post","link":"https:\/\/www.estrategiaconcursos.com.br\/blog\/inequacoes-2-grau\/","title":{"rendered":"Inequa\u00e7\u00f5es do 2\u00ba Grau"},"content":{"rendered":"\n<p class=\"has-medium-font-size\">Aprenda os conceitos essenciais sobre inequa\u00e7\u00f5es do 2\u00ba grau com um resumo para as principais provas de concursos.<\/p>\n\n\n\n<p>Ol\u00e1, pessoal! Tudo bem com voc\u00eas?<\/p>\n\n\n\n<p>As <strong>inequa\u00e7\u00f5es do 2\u00ba grau<\/strong> s\u00e3o uma extens\u00e3o das equa\u00e7\u00f5es do 2\u00ba grau e aparecem com muita frequ\u00eancia em concursos como <strong><\/strong><strong><a href=\"https:\/\/conhecimento.fgv.br\/concursos\" target=\"_blank\" rel=\"noreferrer noopener\">FGV<\/a>, <a href=\"https:\/\/www.cebraspe.org.br\/concursos\/\" target=\"_blank\" rel=\"noreferrer noopener\">Cebraspe<\/a>, <a href=\"https:\/\/www.concursosfcc.com.br\/\" target=\"_blank\" rel=\"noreferrer noopener\">FCC<\/a><\/strong> e <a href=\"https:\/\/www.quadrix.org.br\/\" target=\"_blank\" rel=\"noreferrer noopener\">Quadrix<\/a>.<br \/>Ao contr\u00e1rio das equa\u00e7\u00f5es, que buscam os valores exatos de <strong>x<\/strong>, as inequa\u00e7\u00f5es procuram <strong>intervalos<\/strong> em que a express\u00e3o quadr\u00e1tica \u00e9 <strong>positiva<\/strong>, <strong>negativa<\/strong>, <strong>n\u00e3o positiva<\/strong> ou <strong>n\u00e3o negativa<\/strong>.<\/p>\n\n\n\n<p>Neste artigo, vamos entender como funcionam as inequa\u00e7\u00f5es do 2\u00ba grau, com explica\u00e7\u00f5es e exemplos que facilitam a compreens\u00e3o, para que voc\u00ea possa revisar a mat\u00e9ria de forma r\u00e1pida e estrat\u00e9gica.<\/p>\n\n\n\n<p><strong>Confira os t\u00f3picos que ser\u00e3o abordados:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>O que s\u00e3o inequa\u00e7\u00f5es do 2\u00ba grau;<\/li>\n\n\n\n<li>Tipos de inequa\u00e7\u00f5es do 2\u00ba grau;<\/li>\n\n\n\n<li>Como resolver inequa\u00e7\u00f5es do 2\u00ba grau;<\/li>\n\n\n\n<li>Exerc\u00edcio resolvido;<\/li>\n\n\n\n<li>Resumo.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading has-cyan-bluish-gray-background-color has-background\" id=\"h-o-que-sao-inequacoes-do-2\u00ba-grau\">O Que S\u00e3o Inequa\u00e7\u00f5es do 2\u00ba Grau<\/h2>\n\n\n\n<p>As <strong>inequa\u00e7\u00f5es<\/strong> s\u00e3o semelhantes \u00e0s equa\u00e7\u00f5es, mas em vez de igualdade (=), apresentam <strong>desigualdade<\/strong> entre as express\u00f5es. Ou seja, mostram <strong>quando uma express\u00e3o \u00e9 maior, menor, maior ou igual, ou menor ou igual<\/strong> que outra.<\/p>\n\n\n\n<p>Uma <strong>inequa\u00e7\u00e3o do 2\u00ba grau<\/strong> \u00e9 uma desigualdade envolvendo uma express\u00e3o quadr\u00e1tica, que possui um dos seguintes formatos:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>ax\u00b2 + bx + c > 0<\/li>\n\n\n\n<li>ax\u00b2 + bx + c \u2265 0<\/li>\n\n\n\n<li>ax\u00b2 + bx + c &lt; 0<\/li>\n\n\n\n<li>ax\u00b2 + bx + c \u2264 0<\/li>\n<\/ul>\n\n\n\n<p>Os s\u00edmbolos usados s\u00e3o:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>S\u00edmbolo<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Leitura<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&gt;&nbsp;<\/td><td class=\"has-text-align-center\" data-align=\"center\">maior que<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&lt;&nbsp;<\/td><td class=\"has-text-align-center\" data-align=\"center\">menor que<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\u2265<\/td><td class=\"has-text-align-center\" data-align=\"center\">maior ou igual a<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\u2264<\/td><td class=\"has-text-align-center\" data-align=\"center\">menor ou igual a<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Exemplos de inequa\u00e7\u00f5es do 2\u00ba grau:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u00b2 &#8211; 4x + 3 > 0<\/li>\n\n\n\n<li>2x\u00b2 + 5x \u2265 3<\/li>\n\n\n\n<li>-x\u00b2 + 9 &lt; 0<\/li>\n<\/ul>\n\n\n\n<p>Assim como nas equa\u00e7\u00f5es de 2\u00ba grau, devemos encontrar a forma reduzida:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>ax\u00b2 + bx + c<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-cyan-bluish-gray-background-color has-background\" id=\"h-tipos-de-inequacoes-do-2\u00ba-grau\">Tipos de Inequa\u00e7\u00f5es do 2\u00ba Grau<\/h2>\n\n\n\n<p>As inequa\u00e7\u00f5es podem ser classificadas de acordo com o sinal do coeficiente <strong>a<\/strong>:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>Tipo<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Sinal de \u201ca\u201d<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Forma da par\u00e1bola<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Tend\u00eancia<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>Concavidade para cima<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">(a &gt; 0)<\/td><td class=\"has-text-align-center\" data-align=\"center\">U<\/td><td class=\"has-text-align-center\" data-align=\"center\">A express\u00e3o tende a +\u221e<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>Concavidade para baixo<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">(a &lt; 0)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u2229<\/td><td class=\"has-text-align-center\" data-align=\"center\">A express\u00e3o tende a \u2212\u221e<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Al\u00e9m disso, a natureza das ra\u00edzes da equa\u00e7\u00e3o associada <strong>ax\u00b2 + bx + c = 0<\/strong> determina a resposta.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>Discriminante (\u0394)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Ra\u00edzes<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>Efeito na inequa\u00e7\u00e3o<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\u0394 &gt; 0<\/td><td class=\"has-text-align-center\" data-align=\"center\">2 ra\u00edzes reais distintas<\/td><td class=\"has-text-align-center\" data-align=\"center\">A par\u00e1bola cruza o eixo x \u2192 muda de sinal<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\u0394 = 0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1 raiz real (dupla)<\/td><td class=\"has-text-align-center\" data-align=\"center\">A par\u00e1bola toca o eixo x \u2192 o sinal s\u00f3 muda no ponto<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\u0394 &lt; 0<\/td><td class=\"has-text-align-center\" data-align=\"center\">Sem ra\u00edzes reais<\/td><td class=\"has-text-align-center\" data-align=\"center\">A par\u00e1bola n\u00e3o cruza o eixo x \u2192 sinal \u00e9 <strong>constante<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Essas informa\u00e7\u00f5es s\u00e3o essenciais para o <strong>estudo do sinal<\/strong>, principal m\u00e9todo utilizado.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-cyan-bluish-gray-background-color has-background\" id=\"h-como-resolver-inequacoes-do-2\u00ba-grau\">Como Resolver Inequa\u00e7\u00f5es do 2\u00ba Grau<\/h2>\n\n\n\n<p>O m\u00e9todo mais usado para resolver inequa\u00e7\u00f5es do 2\u00ba grau em provas \u00e9 o <strong>estudo do sinal da fun\u00e7\u00e3o quadr\u00e1tica<\/strong>.<\/p>\n\n\n\n<p><strong>1\u00ba Passo) Transformar a inequa\u00e7\u00e3o em equa\u00e7\u00e3o e encontre as ra\u00edzes<\/strong><\/p>\n\n\n\n<p>Resolva:<\/p>\n\n\n\n<p class=\"has-text-align-center\">ax\u00b2 + bx + c = 0<\/p>\n\n\n\n<p>para encontrar as ra\u00edzes.<\/p>\n\n\n\n<p><strong>2\u00ba Passo) Identificar a concavidade<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Se a > 0, concavidade para <strong>cima<\/strong> &#8211; <strong>U <\/strong>&#8211; positiva fora das ra\u00edzes;<\/li>\n\n\n\n<li>Se a &lt; 0, concavidade para <strong>baixo <\/strong>&#8211;<strong>\u2229<\/strong> &#8211; positiva entre as ra\u00edzes.<\/li>\n<\/ul>\n\n\n\n<p><strong>3\u00ba Passo) Construir a reta de sinais (estudo do sinal)<\/strong><\/p>\n\n\n\n<p>Coloque as ra\u00edzes encontradas na reta num\u00e9rica e analise os intervalos.<\/p>\n\n\n\n<p><strong>4\u00ba Passo) Escolher os intervalos desejados<\/strong><\/p>\n\n\n\n<p>De acordo com a desigualdade:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Maior que zero <\/strong>(> 0) \u2192 escolher regi\u00f5es <strong>positivas.<\/strong><\/li>\n\n\n\n<li><strong>Menor que zero<\/strong> (&lt; 0) \u2192 escolher regi\u00f5es <strong>negativas.<\/strong><\/li>\n\n\n\n<li><strong>Com igualdade<\/strong> (\u2265 ou \u2264) \u2192 incluir as ra\u00edzes.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading has-cyan-bluish-gray-background-color has-background\" id=\"h-exercicio-resolvido\">Exerc\u00edcio Resolvido<\/h2>\n\n\n\n<p><strong>(VUNESP &#8211; Prefeitura de Rio Claro &#8211; Professor de Educa\u00e7\u00e3o B\u00e1sica II \u2013 2022) <\/strong>A solu\u00e7\u00e3o da inequa\u00e7\u00e3o do 2\u00ba grau \u2013x<sup>2<\/sup> \u2013 3x \u2264 \u201310 \u00e9 corretamente apresentada na alternativa:<\/p>\n\n\n\n<p>a) [\u2013\u221e, \u20135] \u222a [2, +\u221e]<br \/>b) [\u2013\u221e, \u20132] \u222a [5, +\u221e]<br \/>c) [\u20135, 2]<br \/>d) [\u20132, 5]<br \/>e) [\u20133, 10]<\/p>\n\n\n\n<p><strong>1) Encontrando as ra\u00edzes da equa\u00e7\u00e3o associada:<\/strong><\/p>\n\n\n\n<p>\u2013x<sup>2<\/sup> \u2013 3x \u2264 \u201310<br \/>\u2013x<sup>2<\/sup> \u2013 3x + 10 \u2264 0<br \/>-x\u00b2 &#8211; 3x + 10 = 0 \u2192 <strong>a = -1; b = -3; c = 10<\/strong><\/p>\n\n\n\n<p>Utilizando a f\u00f3rmula de Bhaskara, chegamos em:<\/p>\n\n\n\n<p>\u0394 = (-3)\u00b2 &#8211; 4.(-1).10 = 49<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"261\" height=\"135\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01190946\/image-5.png\" alt=\"F\u00f3rmula de Bhaskara - Inequa\u00e7\u00f5es do 2\u00ba Grau\" class=\"wp-image-1678257\" srcset=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01190946\/image-5.png 261w, https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01190946\/image-5-150x78.png 150w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/figure>\n\n\n\n<p><strong>2) Identificando a concavidade:<\/strong><\/p>\n\n\n\n<p>Como a = -1 &lt; 0, a par\u00e1bola possui concavidade para baixo (\u2229).<\/p>\n\n\n\n<p><strong>3) Estudo do sinal:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"763\" height=\"344\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01185906\/image-4.png\" alt=\"Estudo do Sinal - Inequa\u00e7\u00f5es do 2\u00ba Grau\" class=\"wp-image-1678245\" style=\"width:581px;height:auto\" srcset=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01185906\/image-4.png 763w, https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01185906\/image-4-300x135.png 300w, https:\/\/dhg1h5j42swfq.cloudfront.net\/2025\/12\/01185906\/image-4-150x68.png 150w\" sizes=\"auto, (max-width: 763px) 100vw, 763px\" \/><\/figure>\n\n\n\n<p><strong>4) Escolha do intervalo:<\/strong><\/p>\n\n\n\n<p>Como queremos a parte negativa do intervalo (\u2013x<sup>2<\/sup> \u2013 3x + 10 \u2264 0), devemos escolher os valores menores que -5 e os maiores que 2. Al\u00e9m disso, como a desigualdade \u00e9 menor ou igual (\u2264), tamb\u00e9m devemos incluir as ra\u00edzes -5 e 2.<\/p>\n\n\n\n<p>Portanto, a resposta correta \u00e9 a alternativa \u201c<strong>a<\/strong>\u201d: <strong>[\u2013\u221e, \u20135] <\/strong><strong>\u222a<\/strong><strong> [2, +\u221e]<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-cyan-bluish-gray-background-color has-background\" id=\"h-finalizando-inequacoes-do-2\u00ba-grau\">Finalizando \u2013 Inequa\u00e7\u00f5es do 2\u00ba Grau<\/h2>\n\n\n\n<p>As <strong>inequa\u00e7\u00f5es do 2\u00ba grau<\/strong> aparecem frequentemente em provas de concursos, e dominar o <strong>estudo do sinal<\/strong> \u00e9 essencial para resolv\u00ea-las com rapidez e seguran\u00e7a. O m\u00e9todo \u00e9 simples: basta encontrar as ra\u00edzes, analisar a concavidade e identificar os intervalos adequados. A chave para o dom\u00ednio est\u00e1 na <strong>pr\u00e1tica<\/strong>.<\/p>\n\n\n\n<p>\u00c9 importante refor\u00e7ar que este conte\u00fado <strong>deve ser utilizado como complemento ao material em PDF<\/strong>, onde a abordagem \u00e9 aprofundada e completa. Al\u00e9m disso, \u00e9 fundamental praticar com muitas quest\u00f5es, preferencialmente separadas por banca, para entender as diferentes formas de cobran\u00e7a.<\/p>\n\n\n\n<p>Quer se aprofundar no tema? O Estrat\u00e9gia Concursos disponibiliza <strong>materiais em PDF completos e atualizados<\/strong>, com teoria detalhada, quest\u00f5es comentadas e videoaulas direcionadas para cada concurso. Com pr\u00e1tica e um bom material, voc\u00ea estar\u00e1 preparado para resolver qualquer quest\u00e3o de inequa\u00e7\u00f5es do 2\u00ba grau que aparecer na sua prova.<\/p>\n\n\n\n<p>Acesse <a href=\"https:\/\/www.estrategiaconcursos.com.br\/\">os cursos do Estrat\u00e9gia Concursos<\/a> e fortale\u00e7a sua prepara\u00e7\u00e3o com um conte\u00fado de alto n\u00edvel.<\/p>\n\n\n\n<p>Bons estudos e at\u00e9 a pr\u00f3xima!<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\" id=\"cursos-assinatura\">Cursos e Assinaturas<\/h2>\n\n\n\n<p>Prepare-se com o melhor material e com quem mais aprova em Concursos P\u00fablicos em todo o pa\u00eds!<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-cyan-blue-color has-text-color has-background\" style=\"background-color:#0078bd\"><a href=\"https:\/\/www.estrategiaconcursos.com.br\/blog\/concursos-abertos\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-white-color\">Concursos Abertos<\/mark><\/strong><\/a><\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-cyan-blue-color has-text-color has-background\" style=\"background-color:#0078bd\"><a href=\"https:\/\/www.estrategiaconcursos.com.br\/blog\/concursos-2025\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-white-color\">Concursos 2025<\/mark><\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aprenda os conceitos essenciais sobre inequa\u00e7\u00f5es do 2\u00ba grau com um resumo para as principais provas de concursos. Ol\u00e1, pessoal! Tudo bem com voc\u00eas? As inequa\u00e7\u00f5es do 2\u00ba grau s\u00e3o uma extens\u00e3o das equa\u00e7\u00f5es do 2\u00ba grau e aparecem com muita frequ\u00eancia em concursos como FGV, Cebraspe, FCC e Quadrix.Ao contr\u00e1rio das equa\u00e7\u00f5es, que buscam [&hellip;]<\/p>\n","protected":false},"author":3098,"featured_media":1622782,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"post_tipo":"article","footnotes":""},"categories":[2313,1,220180,3766,220184,220242,214992,220181,1983,220182,220183,1988,222278,220191,1984,220737,214710,222239,222235,222236,222237],"tags":[],"tax_estado":[219964,219965,219966,219967,219968,219969,219963,219970,219971,219972,219973,219974,219975,219976,219977,219978,219979,219980,219981,219982,219983,219984,219985,219986,219987,219988,219989,219990],"class_list":["post-1678234","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-carreiras-bancarias","category-cursos-e-concursos","category-controle-gestao","category-educacional","category-engenharias-e-ti","category-estrategia-militares","category-concursos-administrativos","category-area-financeira","category-fiscal-estadual","category-fiscal-federal","category-fiscal-municipal","category-policia","category-prefeituras","category-previdenciaria","category-tribunais","category-tribunais-tribunais","category-tribunais-de-contas","category-tribunais-regionais-federais","category-tribunal-de-justica","category-tribunal-regional-do-trabalho","category-tribunal-regional-eleitoral","tax_estado-ac","tax_estado-al","tax_estado-am","tax_estado-ap","tax_estado-ba","tax_estado-ce","tax_estado-concursos-federais","tax_estado-df","tax_estado-es","tax_estado-go","tax_estado-ma","tax_estado-mg","tax_estado-ms","tax_estado-mt","tax_estado-pa","tax_estado-pb","tax_estado-pe","tax_estado-pi","tax_estado-pr","tax_estado-rj","tax_estado-rn","tax_estado-ro","tax_estado-rr","tax_estado-rs","tax_estado-sc","tax_estado-se","tax_estado-sp","tax_estado-to"],"acf":[],"yoast_head":"<!-- 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