{"id":101826,"date":"2023-09-30T07:00:00","date_gmt":"2023-09-30T10:00:00","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=101826"},"modified":"2023-10-03T14:40:52","modified_gmt":"2023-10-03T17:40:52","slug":"equacoes-2","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/","title":{"rendered":"Equa\u00e7\u00f5es: principais tipos, f\u00f3rmulas e aplica\u00e7\u00f5es"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Voc&ecirc; j&aacute; viu alguma quest&atilde;o sobre <span id=\"urn:enhancement-85fe0264-ac65-4700-ba3b-f716d9865f78\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> no <span id=\"urn:enhancement-52bcb251-2831-4ba2-a5e8-768dfc8e4fa6\" class=\"textannotation disambiguated wl-event\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/enem-64632\">Enem<\/span>? O tema, que &eacute; muito abrangente, aparece das mais diversas formas. Seja por meio de problemas l&oacute;gicos, apresenta&ccedil;&atilde;o de gr&aacute;ficos e tabelas ou utiliza&ccedil;&atilde;o de f&oacute;rmulas, os c&aacute;lculos de <span id=\"urn:enhancement-b8613015-278c-4431-8b46-2ddcc87d3bff\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> est&atilde;o entre os mais cobrados na prova de <span id=\"urn:enhancement-b46642a9-951d-49f3-b307-4138158b40a5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>.<p class=\"wp-block-paragraph\">Neste artigo, voc&ecirc; encontra um panorama geral sobre o tema em seus diferentes aspectos: <span id=\"urn:enhancement-610ccad6-987b-4b81-ad6d-f4ed5cee8045\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> de primeiro e segundo grau, alg&eacute;brica, modulares, logar&iacute;tmica, trigonom&eacute;trica, exponencial, irracional, com suas f&oacute;rmulas e principais aplica&ccedil;&otilde;es. Acompanhe agora!<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#O-que-sao-equacoes-algebricas\" >O que s&atilde;o equa&ccedil;&otilde;es alg&eacute;bricas?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-do-primeiro-grau\" >Equa&ccedil;&atilde;o do primeiro grau<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-do-segundo-grau\" >Equa&ccedil;&atilde;o do segundo grau<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-biquadrada\" >Equa&ccedil;&atilde;o biquadrada<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-modular\" >Equa&ccedil;&atilde;o modular<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-exponencial\" >Equa&ccedil;&atilde;o exponencial<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-logaritmica\" >Equa&ccedil;&atilde;o logar&iacute;tmica&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Equacao-trigonometrica\" >Equa&ccedil;&atilde;o trigonom&eacute;trica<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Resolucao-de-equacoes\" >Resolu&ccedil;&atilde;o de equa&ccedil;&otilde;es<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Questoes-de-equacoes-no-Enem\" >Quest&otilde;es de equa&ccedil;&otilde;es no Enem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes-2\/#Estude-mais-com-a-Coruja\" >Estude mais com a Coruja!<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-equacoes-algebricas\"><span class=\"ez-toc-section\" id=\"O-que-sao-equacoes-algebricas\"><\/span>O que s&atilde;o equa&ccedil;&otilde;es alg&eacute;bricas?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As <span id=\"urn:enhancement-967159bd-1ad8-407f-8079-5d0281861b61\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> s&atilde;o ferramentas <span id=\"urn:enhancement-6ab4f500-8c0a-421a-a27b-cb80827feab2\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> em que h&aacute; uma igualdade entre valores e opera&ccedil;&otilde;es, entretanto, um dos termos dentro do c&aacute;lculo n&atilde;o &eacute; conhecido, ele &eacute; chamado de <strong>inc&oacute;gnita<\/strong>. Diante disso, &eacute; necess&aacute;rio realizar manipula&ccedil;&otilde;es <span id=\"urn:enhancement-93829e0a-d5c8-4e9f-929a-ac9bf4abc5bc\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> para encontrar o valor desconhecido. Veja exemplos:<\/p><p class=\"has-text-align-center wp-block-paragraph\">4x + 12 = 3x + 15<\/p><p class=\"has-text-align-center wp-block-paragraph\">5y &ndash; 3y + 7= y + 8<\/p><p class=\"wp-block-paragraph\">Observe que a inc&oacute;gnita &eacute; descrita com letras e, geralmente, s&atilde;o escolhidas entre x, y ou z. Mas isso n&atilde;o &eacute; uma <span id=\"urn:enhancement-6835388d-158b-41d1-a99f-20aea2e6c2bf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">regra<\/span> e qualquer letra do alfabeto pode ser utilizada para representar o n&uacute;mero desconhecido. Essa jun&ccedil;&atilde;o entre n&uacute;meros e letras, &eacute; denominada de <strong><span id=\"urn:enhancement-c1391001-ca41-4cef-8b52-590103687d54\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/algebra\">&aacute;lgebra<\/span><\/strong>, ent&atilde;o as <span id=\"urn:enhancement-3f126b11-1b2c-4687-a533-a86c42084746\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> s&atilde;o classificadas como alg&eacute;bricas.&nbsp;<\/p><p class=\"wp-block-paragraph\">O objetivo principal na resolu&ccedil;&atilde;o de uma <span id=\"urn:enhancement-908db7ca-ff7a-403c-9573-d25389a62736\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> &eacute;<strong> encontrar um valor <\/strong>para inc&oacute;gnita apresentada. Para isso, &eacute; necess&aacute;rio que o c&aacute;lculo deixe de ser uma <span id=\"urn:enhancement-155e070c-5874-479b-9c9a-f84a3c566751\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> e se torne uma identidade.<\/p><p class=\"wp-block-paragraph\">Na <span id=\"urn:enhancement-6decf724-e2c5-4d0a-9a48-a9fc4dcb3bdd\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>, identidade &eacute; uma <span id=\"urn:enhancement-e47ff8c4-beb2-45fd-ba99-668435b005ad\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> num&eacute;rica em que o valor de um lado &eacute; exatamente igual ao do outro lado. Veja um exemplo:<\/p><p class=\"has-text-align-center wp-block-paragraph\">4x + 12 = 3x + 15<\/p><p class=\"has-text-align-center wp-block-paragraph\">4x &ndash; 3x = 15 &ndash; 12<\/p><p class=\"has-text-align-center wp-block-paragraph\">x = 3<\/p><p class=\"wp-block-paragraph\">Ao fazer a substitui&ccedil;&atilde;o da inc&oacute;gnita pelo seu valor real (x=3), teremos que:<\/p><p class=\"has-text-align-center wp-block-paragraph\">4x + 12 = 3x + 15<\/p><p class=\"has-text-align-center wp-block-paragraph\">4.3 + 12 = 3.3 + 15<\/p><p class=\"has-text-align-center wp-block-paragraph\">24 = 24 (identidade)<\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-do-primeiro-grau\"><span class=\"ez-toc-section\" id=\"Equacao-do-primeiro-grau\"><\/span>Equa&ccedil;&atilde;o do primeiro grau<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As <span id=\"urn:enhancement-0e78021e-fd2a-4737-989c-eadb810dfcdc\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> podem ser classificadas conforme seu <strong>grau<\/strong>. O grau &eacute; dado pelo <strong>maior <span id=\"urn:enhancement-9a15f5bc-815e-43c4-8e7e-a75979e6fb64\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoente<\/span> na inc&oacute;gnita<\/strong>. Perceba, ent&atilde;o, que n&atilde;o importa qual ser&aacute; o <span id=\"urn:enhancement-900a4b00-e061-4162-8cde-57a348341892\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoente<\/span> em outros n&uacute;meros, basta observar os expoentes que elevam a inc&oacute;gnita. Dessa <span id=\"urn:enhancement-9e4232fe-93f8-4aa9-8761-96f85c42d777\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>, uma <span id=\"urn:enhancement-ffe4a927-2b86-43f9-8631-faea28cf5ed3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-do-primeiro-grau\">equa&ccedil;&atilde;o do primeiro grau<\/span> cont&eacute;m todas as inc&oacute;gnitas com<strong> <span id=\"urn:enhancement-1d11ec3b-26e9-4ef8-a1be-651e2e5e3644\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoente<\/span> igual a 1<\/strong>.&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">112x + 14 = 100x + 78<\/p><p class=\"has-text-align-center wp-block-paragraph\">x + 6 = 27<\/p><p class=\"has-text-align-center wp-block-paragraph\">45 &ndash; 7x = 11x<\/p><p class=\"has-text-align-center wp-block-paragraph\">4<sup>2<\/sup> + 3x = x + 1&nbsp;&nbsp;<\/p><p class=\"wp-block-paragraph\">Observe que, no &uacute;ltimo exemplo apresentado, um dos termos est&aacute; elevado ao quadrado. Entretanto, n&atilde;o existe pot&ecirc;ncia de 2 que eleve a inc&oacute;gnita em <span id=\"urn:enhancement-b955b9cf-83f1-49da-9c1f-de86e8c053b9\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sistema-internacional-de-unidades\">si<\/span>, por isso a classifica&ccedil;&atilde;o continua sendo de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-do-primeiro-grau\/\" target=\"_blank\">equa&ccedil;&atilde;o do primeiro grau<\/a>.&nbsp;<\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-do-segundo-grau\"><span class=\"ez-toc-section\" id=\"Equacao-do-segundo-grau\"><\/span>Equa&ccedil;&atilde;o do segundo grau<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">No caso da<a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-do-segundo-grau\/\" target=\"_blank\"> equa&ccedil;&atilde;o de segundo grau<\/a>, uma das inc&oacute;gnitas deve estar <span id=\"urn:enhancement-ddf653bc-a57f-4fc2-9ce5-6a8591997623\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">elevada<\/span> ao quadrado (x<sup>2<\/sup>) ou estar multiplicada por outra inc&oacute;gnita (x.y). Nesse caso, existem dois resultados poss&iacute;veis para a mesma inc&oacute;gnita.<\/p><p class=\"wp-block-paragraph\">Isso acontece porque o <strong>n&uacute;mero de solu&ccedil;&otilde;es para uma <span id=\"urn:enhancement-29019928-2713-4dde-ab3a-3f9f25f5725c\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> &eacute; igual ao n&uacute;mero do grau do c&aacute;lculo<\/strong>. Na <span id=\"urn:enhancement-c7c0f7c1-b121-4982-8cca-799e9202e75e\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> de grau 2, dois valores podem ser substitu&iacute;dos na <span id=\"urn:enhancement-24b1ccda-ae75-410c-b00d-7acebd657f21\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> e formar uma identidade. Eles s&atilde;o representados por<strong> x<\/strong><strong><sub>1<\/sub><\/strong> e <strong>x<\/strong><strong><sub>2<\/sub><\/strong>, geralmente.&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">16x &ndash; x<sup>2<\/sup> = 60&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">x<sub>1 <\/sub>= 6&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">x<sub>2 <\/sub>= 10<\/p><p class=\"wp-block-paragraph\">Dada a <span id=\"urn:enhancement-f163fe92-c181-462f-986b-17e8b35647a5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-do-segundo-grau\">equa&ccedil;&atilde;o do segundo grau<\/span> e seus dois resultados, vamos testar se formam identidades se x for substitu&iacute;do por x<sub>1<\/sub> ou x<sub>2<\/sub>.<\/p><p class=\"wp-block-paragraph\">Para x<sub>1<\/sub>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">16x &ndash; x<sup>2<\/sup> = 60<\/p><p class=\"has-text-align-center wp-block-paragraph\">16.6 &ndash; 36 = 60<\/p><p class=\"has-text-align-center wp-block-paragraph\">16.6 = 60 + 36<\/p><p class=\"has-text-align-center wp-block-paragraph\">96 = 96 (&eacute; uma identidade, ent&atilde;o x<sub>1 <\/sub>&eacute; verdadeiro)<\/p><p class=\"wp-block-paragraph\">Para x<sub>2<\/sub>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">16x &ndash; x<sup>2<\/sup> = 60<\/p><p class=\"has-text-align-center wp-block-paragraph\">16.10 &ndash; 100= 60<\/p><p class=\"has-text-align-center wp-block-paragraph\">16.10 = 60 + 100<\/p><p class=\"has-text-align-center wp-block-paragraph\">160 = 160 (tamb&eacute;m &eacute; uma identidade, x<sub>2 <\/sub>&eacute; verdadeiro)<\/p><p class=\"wp-block-paragraph\">A estrutura geral de uma <span id=\"urn:enhancement-912c31ec-53aa-43bd-90e0-3a47d2ffbeac\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> de segundo grau &eacute; dada por:<\/p><p class=\"has-text-align-center wp-block-paragraph\">ax<sup>2<\/sup> + bx + c = y<\/p><p class=\"has-text-align-center wp-block-paragraph\">a,b e c s&atilde;o <span id=\"urn:enhancement-b89b3b47-f653-4755-805d-3060ac1b4cda\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> que regem a inc&oacute;gnita x, resultando no valor y.&nbsp;<\/p><p class=\"wp-block-paragraph\">Note que a, b e c podem ser substitu&iacute;dos por qualquer n&uacute;mero para constituir essa <span id=\"urn:enhancement-6fb4f0cf-0ab7-460b-b364-c6c4f81f79fd\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-do-segundo-grau\">equa&ccedil;&atilde;o do segundo grau<\/span>, como demonstrado abaixo:<\/p><p class=\"has-text-align-center wp-block-paragraph\">4x<sup>2<\/sup> + 3x + 1 = y<\/p><p class=\"has-text-align-center wp-block-paragraph\">a = 4<\/p><p class=\"has-text-align-center wp-block-paragraph\">b = 3<\/p><p class=\"has-text-align-center wp-block-paragraph\">c = 1<\/p><p class=\"wp-block-paragraph\">Para encontrar o valor da inc&oacute;gnita &eacute; poss&iacute;vel utilizar duas f&oacute;rmulas principais, que utilizam os <span id=\"urn:enhancement-9cb7610c-c7dd-474b-a37c-32a3c6607e19\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> como base:<\/p><ul class=\"wp-block-list\">\n<li><strong>Bhaskara<\/strong>: <img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/sGJx8wroLSQ_ys9PIFcOUrK7UHe8wBwJZMKXk8YG6it8rhr1Ll2xH_YSjvyIR8oKE6FMwtBDaCBNjQ0CwMSW5Op5034uSR7vNLu4vbZ71NuGqhvSMM3hR22tBZBy7Vk8H_eur0Dr0BtM9Ey1QoCd9Pg\" width=\"155\" height=\"48\"> ou;<\/li>\n\n\n\n<li><strong>Soma e produto<\/strong>: x<sub>1<\/sub>.x<sub>2<\/sub> = c\/a e x<sub>1<\/sub> + x<sub>2<\/sub> = -b\/a<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-equacao-biquadrada\"><span class=\"ez-toc-section\" id=\"Equacao-biquadrada\"><\/span>Equa&ccedil;&atilde;o biquadrada<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Uma <span id=\"urn:enhancement-0b3a8485-8586-4159-9ecd-c88efed62b41\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> biquadrada possui pelo menos um <span id=\"urn:enhancement-75c4ba83-63b3-40f2-8e86-a9a717244169\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span> elevado ao <strong>quarto grau<\/strong>, sua lei de forma&ccedil;&atilde;o geral &eacute; dada por:<\/p><p class=\"has-text-align-center wp-block-paragraph\">ax<sup>4<\/sup> + bx<sup>2<\/sup> + c = 0<\/p><p class=\"has-text-align-center wp-block-paragraph\">a, b e c podem ser qualquer valor real diferente de zero.<\/p><p class=\"wp-block-paragraph\">Note como as <strong>inc&oacute;gnitas est&atilde;o elevadas a expoentes pares <\/strong>(x<sup>4 <\/sup>e x<sup>2<\/sup>), essa pode ser uma caracter&iacute;stica importante para reconhecer a <span id=\"urn:enhancement-ed9bc088-763b-471f-afea-ba60070ddaf2\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> biquadrada em quest&otilde;es de prova. Por essa propriedade, ela &eacute; considerada uma <span id=\"urn:enhancement-0136e4c0-de3b-422a-b332-0fee1be2e617\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> redut&iacute;vel ao segundo grau, o que facilita os c&aacute;lculos. Veja abaixo:<\/p><p class=\"has-text-align-center wp-block-paragraph\">ax<sup>4<\/sup> + bx<sup>2<\/sup> + c = 0<\/p><p class=\"has-text-align-center wp-block-paragraph\">a.x<sup>2<\/sup>.x<sup>2<\/sup> + b.x<sup>2<\/sup> + c = 0<\/p><p class=\"wp-block-paragraph\">Para a transforma&ccedil;&atilde;o, considera-se que x<sup>2<\/sup>&nbsp; = y, ent&atilde;o a <span id=\"urn:enhancement-04ca98c2-2f40-4aa8-bb3f-f76e6d8298f3\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> passa a ser<\/p><p class=\"has-text-align-center wp-block-paragraph\">a.y.y + b.y + c = 0<\/p><p class=\"has-text-align-center wp-block-paragraph\">ay<sup>2<\/sup> + by + c = 0<\/p><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-19be18d1-46a9-4bac-8ea7-88eb2e784a9d\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> biquadrada de inc&oacute;gnita x transformou-se em uma <span id=\"urn:enhancement-aa423773-9380-4ae8-a9a0-0f5656513e84\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> de segundo grau de inc&oacute;gnita y. Ao final, encontram-se y<sub>1<\/sub> e y<sub>2<\/sub>. Cada um desses valores devem ser igualados a x<sup>2<\/sup> e formar um novo c&aacute;lculo do segundo grau, com duas respostas cada (x<sub>1<\/sub>, x<sub>2<\/sub>, x<sub>3<\/sub>, x<sub>4<\/sub>).<\/p><p class=\"has-text-align-center wp-block-paragraph\">y<sub>1<\/sub> = x<sup>2<\/sup> &rArr; x<sub>1 <\/sub>e x<sub>2<\/sub><\/p><p class=\"has-text-align-center wp-block-paragraph\">y<sub>2<\/sub> = x<sup>2<\/sup> &rArr; x<sub>3&nbsp; <\/sub>e x<sub>4<\/sub><\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-modular\"><span class=\"ez-toc-section\" id=\"Equacao-modular\"><\/span>Equa&ccedil;&atilde;o modular<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">O m&oacute;dulo, na <span id=\"urn:enhancement-afca3195-86c4-470c-9abb-22d37720de80\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>, fornece o valor absoluto de algo, ou seja, a quantidade ser&aacute; sempre contada em n&uacute;meros positivos. Ent&atilde;o, uma <a href=\"https:\/\/www.youtube.com\/watch?v=L_kHLil45kQ&amp;pp=ygUZZXN0cmF0ZWdpYSB2ZXN0IGVxdWHDp8Ojbw%3D%3D\" target=\"_blank\">equa&ccedil;&atilde;o modular <\/a>possui pelo menos uma inc&oacute;gnita em valor absoluto.Ou seja, &eacute; necess&aacute;rio que a inc&oacute;gnita apare&ccedil;a pelo menos uma vez dentro do m&oacute;dulo (lxl).&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">lxl = x<\/p><p class=\"has-text-align-center wp-block-paragraph\">l-xl = x<\/p><p class=\"wp-block-paragraph\">No exemplo acima fica evidente que a modula&ccedil;&atilde;o altera completamente os resultados da <span id=\"urn:enhancement-f458f3be-d21f-418b-b74e-1ff4cd467dcb\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, por isso &eacute; importante atentar-se a essa entidade da <span id=\"urn:enhancement-259e682d-622d-4805-9739-245710322e52\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>. Em alguns casos, ainda, ocorrem uma opera&ccedil;&otilde;es dentro do m&oacute;dulo, ent&atilde;o o resultado da conta &eacute; quem deve aparecer em valor absoluto. Seja x<sub> <\/sub>= -7, a <span id=\"urn:enhancement-ebbfcdbe-e6ef-423e-a5f1-5c31ad5fd03d\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> abaixo deve ser resolvida da seguinte <span id=\"urn:enhancement-05a62f20-a811-4b41-a1ef-1cfa601ec7ac\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">lx + 4l =&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">l- 7 + 4l =&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">l-3l =&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">l-3l = 3<\/p><p class=\"has-text-align-center wp-block-paragraph\">lx + 4l = 3<\/p><p class=\"wp-block-paragraph\">Um dos pontos mais importantes das <span id=\"urn:enhancement-a45f6e4d-9e67-4994-aa11-076d934e55eb\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> modulares est&aacute; na sua tend&ecirc;ncia para a positividade. Quando uma <span id=\"urn:enhancement-02bfc8ab-734c-4eca-9194-eaa41f288141\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> modular tem rela&ccedil;&atilde;o com uma <a href=\"http:\/\/panorama%20geral%20sobre%20o%20tema%20em%20seus%20diferentes%20aspectos:%20equa%C3%A7%C3%B5es%20primeiro%20e%20segundo%20grau,%20alg%C3%A9brica,%20modulares,%20logar%C3%ADtmica,%20trigonom%C3%A9trica,%20exponencial,%20irracional,%20com%20suas%20f%C3%B3rmulas%20e%20principais%20aplica%C3%A7%C3%B5e\" target=\"_blank\">fun&ccedil;&atilde;o matem&aacute;tica<\/a> f(x), observa-se que o gr&aacute;fico fica tendencioso para y&gt;0, observe a figura:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/X24nF3ZRyL7a-fqp_99iBEnF7oSEp2YloRGEY0R2wKi8oKQ1VUoFzp2QkEHTs1ww9jjvzkAChsK6DtJlMWki-IDKnIBUPCd5CEu0-s_vQF6trEhd80e_sdx-HTFoHqrbsIcoUZVlfyCC5cFAp7bYUIs\" alt=\"equa&ccedil;&atilde;o modular\" width=\"604\" height=\"331\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Essa imagem &eacute; uma representa&ccedil;&atilde;o dos tipos mais cl&aacute;ssicos de <span id=\"urn:enhancement-52da457a-dfc1-4131-aabf-4d469ca87cca\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> modular, mas n&atilde;o &eacute; uma <span id=\"urn:enhancement-d02c11d5-59a0-4531-87e4-a42171275585\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">regra<\/span> que y&gt;0 para que a classifica&ccedil;&atilde;o seja v&aacute;lida.<\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-exponencial\"><span class=\"ez-toc-section\" id=\"Equacao-exponencial\"><\/span>Equa&ccedil;&atilde;o exponencial<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Nas <span id=\"urn:enhancement-c8f86718-7518-4e64-a5ae-8f44a8fd1236\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> exponenciais, pelo menos uma inc&oacute;gnita deve aparecer no <span id=\"urn:enhancement-338defd0-a575-45fd-bde7-f1bf0da2d978\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoente<\/span> de um n&uacute;mero. Com isso, muitas vezes, &eacute; necess&aacute;rio realizar a opera&ccedil;&atilde;o inversa (logaritmo) para encontrar a resolu&ccedil;&atilde;o da <span id=\"urn:enhancement-4a2879ae-ffbf-4cf7-bbec-056dafeb430e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o alg&eacute;brica<\/span>.&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">4<sup>x<\/sup> = 25 &ndash; 9&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">5<sup>x<\/sup> = 101 + 8.x<\/p><p class=\"wp-block-paragraph\">Observe que a inc&oacute;gnita pode aparecer como um termo simples da <span id=\"urn:enhancement-57fab490-81ce-4e02-8b07-45a3fbaff2d9\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> e tamb&eacute;m como <span id=\"urn:enhancement-b5699851-46c6-4366-aab7-1075d9f7c6e1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoente<\/span> de outro valor. Al&eacute;m disso, &eacute; importante conhecer as principais <span id=\"urn:enhancement-d5d90764-e3d9-4b02-ab65-8f6afcce35f5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">pot&ecirc;ncias<\/span> e seus resultados, para facilitar a resolu&ccedil;&atilde;o de c&aacute;lculos que terminam em valores comuns, como 2<sup>2<\/sup> = 4, 2<sup>3<\/sup> = 8, 5<sup>2<\/sup> = 25, 4<sup>2<\/sup> = 16, entre outros.<\/p><p class=\"wp-block-paragraph\">Acompanhe a solu&ccedil;&atilde;o de uma <a href=\"https:\/\/www.youtube.com\/watch?v=YINqLzgH7GE&amp;pp=ygUZZXN0cmF0ZWdpYSB2ZXN0IGVxdWHDp8Ojbw%3D%3D\" target=\"_blank\">equa&ccedil;&atilde;o exponencial<\/a>:<\/p><p class=\"wp-block-paragraph\">CEFET-MG 2010<\/p><p class=\"wp-block-paragraph\">Considerando a <span id=\"urn:enhancement-8d02d7ff-afe8-47b2-89ac-7b46b7f758ea\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> 2<sup>x <\/sup>= 5 e que log 2 = 0,3, o valor mais pr&oacute;ximo de X &eacute;:<\/p><p class=\"wp-block-paragraph\">a) 2,2<\/p><p class=\"wp-block-paragraph\">b) 2,3<\/p><p class=\"wp-block-paragraph\">c) 2,4<\/p><p class=\"wp-block-paragraph\">d) 2,5<\/p><p class=\"wp-block-paragraph\">2<sup>x<\/sup> = 5<\/p><p class=\"wp-block-paragraph\">log 2<sup>x<\/sup> = log 5<\/p><p class=\"wp-block-paragraph\">A partir das propriedades logar&iacute;tmicas, &eacute; poss&iacute;vel substituir 5 por 10\/2 e admitir que log 2<sup>x<\/sup> = x.log 2:<\/p><p class=\"has-text-align-center wp-block-paragraph\">x. log 2 = log (10\/2)<\/p><p class=\"has-text-align-center wp-block-paragraph\">x .log 2 = log 10 &ndash; log 2<\/p><p class=\"wp-block-paragraph\">Como log 10 = 1 e log 2= 0,3, teremos que:<\/p><p class=\"has-text-align-center wp-block-paragraph\">x.0,3 = 1 &ndash; 0,3<\/p><p class=\"has-text-align-center wp-block-paragraph\">x = 0,7\/03<\/p><p class=\"has-text-align-center wp-block-paragraph\">x=2,33 (alternativa C)<\/p><p class=\"has-text-align-center wp-block-paragraph\">Atente-se para o fato de que as <span id=\"urn:enhancement-1ebc727d-40cc-4647-a342-0d54003c0627\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> logar&iacute;tmica e exponencial se somaram para a resolu&ccedil;&atilde;o dessa quest&atilde;o.<\/p><p class=\"wp-block-paragraph\">+ Veja tamb&eacute;m: <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/equacao-irracional\/\" target=\"_blank\">Equa&ccedil;&atilde;o irracional: o que &eacute; e como resolver<\/a><\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-logaritmica\"><span class=\"ez-toc-section\" id=\"Equacao-logaritmica\"><\/span>Equa&ccedil;&atilde;o logar&iacute;tmica&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">O logaritmo &eacute; uma ferramenta da <span id=\"urn:enhancement-153836b4-c689-4bc6-9f68-ff53e78e4445\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> que permite a manipula&ccedil;&atilde;o dos expoentes em um valor ou inc&oacute;gnita. S&atilde;o a opera&ccedil;&atilde;o inversa da exponencia&ccedil;&atilde;o e s&atilde;o chamados de &ldquo;logs&rdquo; e apresentam um logaritmando e uma base, como mostra a figura a seguir.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/8IHVH7deSP3Qfjn0ead5rRYVmZCiHj7wEZ1gEnzEMJfMgJ3tU3OyEia9LhTDu2SWWA8buZOBJ5TtPEKP9iCSvHIUn5Q53vQPjqd_KO5OSGFMU_ZAuidYIaHX-ES1CVjQv71FLGS0RFiHKlQNDV6HoD0\" alt=\"equa&ccedil;&atilde;o logariimica:  defini&ccedil;&atilde;o de logaritmo\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">b&gt;0, a&gt;0 e a&ne;1<\/p><p class=\"wp-block-paragraph\">Em muitos casos, o exerc&iacute;cio requer que se encontre o valor do logaritmando. Para isso, &eacute; constru&iacute;da uma <span id=\"urn:enhancement-ee57c97f-1649-451c-9a4a-ee80bd04cabe\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> logar&iacute;tmica, como a demonstrada abaixo:<\/p><p class=\"wp-block-paragraph\">log<sub>3 <\/sub>8 + log<sub>3<\/sub>(x+2)= 4<\/p><p class=\"wp-block-paragraph\">Perceba que o logaritmando b = x + 2. Ent&atilde;o, &eacute; necess&aacute;rio aplicar os conhecimentos sobre logaritmos e suas propriedades, de <span id=\"urn:enhancement-7ffd46fb-6fb5-4d8c-847a-8699d558ff21\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> a encontrar o valor da inc&oacute;gnita.<\/p><h2 class=\"wp-block-heading\" id=\"h-equacao-trigonometrica\"><span class=\"ez-toc-section\" id=\"Equacao-trigonometrica\"><\/span>Equa&ccedil;&atilde;o trigonom&eacute;trica<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-d0d2231b-3380-40e9-8f9f-6eb417ecf850\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> trigonom&eacute;trica &eacute; aquela que adiciona os conceitos de grau, seno, <span id=\"urn:enhancement-f5c1dc44-6ea1-41cc-9bea-bdee6adbfeae\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e tangente dos &acirc;ngulos ao contexto da <span id=\"urn:enhancement-c7e3503d-9940-47ce-b29d-587bde321415\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/algebra\">&aacute;lgebra<\/span>. Para resolv&ecirc;-las, &eacute; necess&aacute;rio observar o valor dos <span id=\"urn:enhancement-368ee357-2419-4c2d-8d9d-a1312e4a8949\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> e grandezas trigonom&eacute;tricas e adicionar ao desenvolvimento do c&aacute;lculo. Nesses casos &eacute; relevante lembrar que &#120587; = 180&ordm; e 2&#120587;&nbsp; = 360&ordm;, para facilitar a correla&ccedil;&atilde;o entre radianos e graus.&nbsp;<\/p><p class=\"wp-block-paragraph\">4 sen x = 1<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-e881b026-9297-4316-9e93-b90126ab32a9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span><sup>2<\/sup> x = 1\/2<\/p><p class=\"wp-block-paragraph\">Por conven&ccedil;&atilde;o foram adotados valores de arco para cada um dos &acirc;ngulos e s&atilde;o esses os valores utilizados para a resolu&ccedil;&atilde;o das <span id=\"urn:enhancement-80131a30-cb63-49f0-a268-0cbaa4c5724f\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> trigonom&eacute;tricas. No gr&aacute;fico abaixo, voc&ecirc; pode observar a rela&ccedil;&atilde;o entre seno e <span id=\"urn:enhancement-4cdc1978-c601-4443-961b-55691b2527df\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> com os valores de arco para uma fun&ccedil;&atilde;o vermelha f(x) = sen x e na fun&ccedil;&atilde;o azul f(x) = <span id=\"urn:enhancement-5f966337-49d3-4b5b-9baf-3231b730c4d0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span> (x).<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/muvvdCm4LtnwHmMLUPiJjR4APrk9arwutnDnBlnl1dKFoxhf2K5ngJzW7JPzGWFLQoGylDVahySBBfb5qcxz0XLBaFu5n1t0-8LsGg_5tmiqyd9TNVajjPi4pQxBXA4Sdnijb1XtxzJbihCETRJZ-Ck\" alt=\"gr&aacute;fico de fun&ccedil;&atilde;o trigon&eacute;trica, estudo de equa&ccedil;&otilde;es trigonom&eacute;tricas\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Isso significa, por exemplo, que:<\/p><ul class=\"wp-block-list\">\n<li>seno 0 = 0;<\/li>\n\n\n\n<li>seno 1\/2&#120587; = 1;<\/li>\n\n\n\n<li>seno &#120587; = 0;<\/li>\n\n\n\n<li>seno 3\/2&#120587; = -1;<\/li>\n\n\n\n<li>cosseno 0 = 1;<\/li>\n\n\n\n<li>cosseno 1\/2&#120587; = 0;<\/li>\n\n\n\n<li>cosseno &#120587; = -1; e<\/li>\n\n\n\n<li>cosseno 3\/2&#120587; =0.<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-resolucao-de-equacoes\"><span class=\"ez-toc-section\" id=\"Resolucao-de-equacoes\"><\/span>Resolu&ccedil;&atilde;o de equa&ccedil;&otilde;es<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A resolu&ccedil;&atilde;o de <span id=\"urn:enhancement-d179fd78-2c54-4f6d-b20c-5eae0f7758e0\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> depende de diferentes conhecimentos matem&aacute;ticos. O <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/regra-de-sinais\/\">jogo de sinais<\/a> &eacute; um conhecimento importante, al&eacute;m da manipula&ccedil;&atilde;o de igualdades para que as opera&ccedil;&otilde;es <span id=\"urn:enhancement-062bf63c-8bed-450b-ba2c-1c3f5cdf7853\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> mantenham-se as mesmas no decorrer do c&aacute;lculo.<\/p><p class=\"wp-block-paragraph\">A opera&ccedil;&atilde;o inversa da <span id=\"urn:enhancement-db923991-ec07-4ef1-b12f-444bcc062f73\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> &eacute; a divis&atilde;o, da subtra&ccedil;&atilde;o &eacute; a soma, da exponencia&ccedil;&atilde;o s&atilde;o os logaritmos. Com essas informa&ccedil;&otilde;es, &eacute; poss&iacute;vel inverter os termos de lado na igualdade, preservando a veracidade do c&aacute;lculo.&nbsp;<\/p><p class=\"wp-block-paragraph\">Para al&eacute;m de saber resolver uma <span id=\"urn:enhancement-4d98c8b4-9880-4881-a73c-2d11092c8c40\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, os vestibulares cobram a habilidade de interpretar dados e construir <span id=\"urn:enhancement-8942138c-6670-4aac-a599-45a27bc458bd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es alg&eacute;bricas<\/span> a partir deles. Seja para a constru&ccedil;&atilde;o de gr&aacute;ficos, compreens&atilde;o de lucros e preju&iacute;zos, entre outras problem&aacute;ticas apresentadas em provas.<\/p><h2 class=\"wp-block-heading\" id=\"h-questoes-de-equacoes-no-enem\"><span class=\"ez-toc-section\" id=\"Questoes-de-equacoes-no-Enem\"><\/span>Quest&otilde;es de equa&ccedil;&otilde;es no Enem<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-bc6ebd19-39e5-49e5-be23-a7ccf53d09f0\" class=\"textannotation disambiguated wl-event\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/enem-64632\">ENEM<\/span> 2012<\/p><p class=\"wp-block-paragraph\">Um maquinista de trem ganha R$100,00 por viagem e s&oacute; pode viajar a cada 4 dias. Ele ganha somente se fizer a viagem e sabe que estar&aacute; de f&eacute;rias de 1&ordm; a 10 de junho, quando n&atilde;o poder&aacute; viajar. Sua primeira viagem ocorreu no dia primeiro de janeiro. Considere que o ano tem 365 dias.<\/p><p class=\"wp-block-paragraph\">Se o maquinista quiser ganhar o m&aacute;ximo poss&iacute;vel, quantas viagens precisar&aacute; fazer?<\/p><p class=\"wp-block-paragraph\">a) 37<br>b) 51<br>c) 88<br>d) 89<br>e) 91<\/p><p class=\"wp-block-paragraph\">O racioc&iacute;nio deste exerc&iacute;cio exige a cria&ccedil;&atilde;o de uma <span id=\"urn:enhancement-0fed9945-0c66-4fc0-a9f8-d5db29f46c9a\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> que forne&ccedil;a a quantidade de viagens m&aacute;ximas poss&iacute;veis de serem feitas.<\/p><p class=\"wp-block-paragraph\">O primeiro ponto est&aacute; na observa&ccedil;&atilde;o de que o indiv&iacute;duo possui 365 &ndash; 10 = 355 dias dispon&iacute;veis para o trabalho. Depois disso, nota-se que seu recebimento est&aacute; atrelado a um total de 4 dias trabalhados. De <span id=\"urn:enhancement-ec1ab59d-bb61-48f0-8c7f-16aac449ca93\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> que, o n&uacute;mero de viagens multiplicado pelo n&uacute;mero de dias ativos deve preencher o m&aacute;ximo de tempo do ano:<\/p><p class=\"wp-block-paragraph\">365 &ndash; 10 = 4.x<br>355 = 4.x<br>x = 355\/4<br>x = 88,75<\/p><p class=\"wp-block-paragraph\">Como o n&uacute;mero de viagens n&atilde;o pode ser incompleto, a quantidade m&aacute;xima de trabalhos completos &eacute; de 88, como aponta a alternativa C.<\/p><p class=\"wp-block-paragraph\">Note como a resolu&ccedil;&atilde;o do exerc&iacute;cio agrega a constru&ccedil;&atilde;o de uma <span id=\"urn:enhancement-dcc8555f-8ba3-48ed-a073-19c7f2c4cee1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o alg&eacute;brica<\/span> que transmitisse as ideias enunciadas, de maneira que a <span id=\"urn:enhancement-021dcb9d-1e50-4de1-bc37-8795a0e4e4c0\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> aparece como uma ferramenta de desenvolvimento l&oacute;gico.<\/p><p class=\"wp-block-paragraph\">Alternativa C<\/p><h2 class=\"wp-block-heading\" id=\"h-estude-mais-com-a-coruja\"><span class=\"ez-toc-section\" id=\"Estude-mais-com-a-Coruja\"><\/span>Estude mais com a Coruja!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A banca de provas da sua faculdade dos sonhos tem testes com primeira e segunda fase? O Estrat&eacute;gia Vestibulares pode te ajudar! Nossos simulados s&atilde;o constru&iacute;dos com base no modelo de prova de cada banca, com gabarito e corre&ccedil;&atilde;o, inclusive para os exerc&iacute;cios discursivos da segunda fase.&nbsp;<\/p><p class=\"wp-block-paragraph\">Para estudar e consolidar ainda mais os pontos de maior dificuldade, conte com os cursos da Coruja. Al&eacute;m de aulas claras, did&aacute;ticas e direcionadas, os materiais s&atilde;o acess&iacute;veis em diversas plataformas, com ferramentas interativas, possibilidade de anotar e destacar conte&uacute;dos. Conhe&ccedil;a essa e outras vantagens ao clicar no banner abaixo.<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/EV4.jpg\" alt=\"CTA - Estrat\u00e9gia Vestibulares 4\" title=\"CTA - Estrat\u00e9gia Vestibulares 4\">\n        <\/a>\n\n\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Voc&ecirc; j&aacute; viu alguma quest&atilde;o sobre equa&ccedil;&otilde;es no Enem? O tema, que &eacute; muito abrangente, aparece das mais&hellip;\n","protected":false},"author":23,"featured_media":101818,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-101826","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Equa\u00e7\u00f5es: principais tipos, f\u00f3rmulas e aplica\u00e7\u00f5es<\/title>\n<meta name=\"description\" content=\"Veja um panorama geral sobre: equa\u00e7\u00f5es de primeiro e segundo grau, modulares, logar\u00edtmica, trigonom\u00e9trica, exponencial, irracional e mais!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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