{"id":24625,"date":"2023-03-12T17:39:10","date_gmt":"2023-03-12T17:39:10","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=24625"},"modified":"2023-04-01T14:02:21","modified_gmt":"2023-04-01T13:02:21","slug":"qual-e-o-conjunto-solucao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=24625","title":{"rendered":"Qual \u00e9 o conjunto-solu\u00e7\u00e3o?"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_24625' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_24625' class='GTTabs_curr'><a  id=\"24625_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_24625' ><a  id=\"24625_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_24625'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Qual \u00e9 o conjunto-solu\u00e7\u00e3o de cada uma das seguintes equa\u00e7\u00f5es?<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 160px;\">\n<tbody>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">a)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\({x^2} = 9\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">b)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\({x^2} &#8211; 4 = 0\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">c)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\(4{x^2} &#8211; 25 = 0\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">d)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\(9{x^2} = 16\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">e)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\({x^2} + 1 = 0\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">f)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\( &#8211; 4{x^2} = &#8211; 100\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">g)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\(2{x^2} + 13 = 0\\)<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 3.125%; height: 20px;\">h)<\/td>\n<td style=\"text-align: left; width: 96.7449%; height: 20px;\">\\(64 = {x^2}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_24625' onClick='GTTabs_show(1,24625)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_24625'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>O conjunto-solu\u00e7\u00e3o de cada uma das seguintes equa\u00e7\u00f5es \u00e9 apresentado seguidamente.<\/p>\n<table style=\"border-collapse: collapse;\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td style=\"text-align: left;\">\\({x^2} = 9\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{{x^2} = 9}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\sqrt 9 }&amp; \\vee &amp;{x = + \\sqrt 9 }\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 3}&amp; \\vee &amp;{x = 3}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; 3,\\;3} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>b)<\/td>\n<td style=\"text-align: left;\">\\({x^2} &#8211; 4 = 0\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{{x^2} &#8211; 4 = 0}&amp; \\Leftrightarrow &amp;{{x^2} = 4}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\sqrt 4 }&amp; \\vee &amp;{x = + \\sqrt 4 }\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 2}&amp; \\vee &amp;{x = 2}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; 2,\\;2} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>c)<\/td>\n<td style=\"text-align: left;\">\\(4{x^2} &#8211; 25 = 0\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{4{x^2} &#8211; 25 = 0}&amp; \\Leftrightarrow &amp;{4{x^2} = 25}\\\\{}&amp; \\Leftrightarrow &amp;{{x^2} = \\frac{{25}}{4}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\sqrt {\\frac{{25}}{4}} }&amp; \\vee &amp;{x = + \\sqrt {\\frac{{25}}{4}} }\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\frac{5}{2}}&amp; \\vee &amp;{x = \\frac{5}{2}}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; \\frac{5}{2},\\;\\frac{5}{2}} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>d)<\/td>\n<td style=\"text-align: left;\">\\(9{x^2} = 16\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{9{x^2} = 16}&amp; \\Leftrightarrow &amp;{{x^2} = \\frac{{16}}{9}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\sqrt {\\frac{{16}}{9}} }&amp; \\vee &amp;{x = + \\sqrt {\\frac{{16}}{9}} }\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\frac{4}{3}}&amp; \\vee &amp;{x = \\frac{4}{3}}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; \\frac{4}{3},\\;\\frac{4}{3}} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>e)<\/td>\n<td style=\"text-align: left;\">\\({x^2} + 1 = 0\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{{x^2} + 1 = 0}&amp; \\Leftrightarrow &amp;{{x^2} = &#8211; 1}\\\\{}&amp; \\Leftrightarrow &amp;{x \\in \\emptyset }\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ {} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>f)<\/td>\n<td style=\"text-align: left;\">\\( &#8211; 4{x^2} = &#8211; 100\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{ &#8211; 4{x^2} = &#8211; 100}&amp; \\Leftrightarrow &amp;{{x^2} = 25}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 5}&amp; \\vee &amp;{x = 5}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; 5,\\;5} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>g)<\/td>\n<td style=\"text-align: left;\">\\(2{x^2} + 13 = 0\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{2{x^2} + 13 = 0}&amp; \\Leftrightarrow &amp;{{x^2} = &#8211; \\frac{{13}}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{x \\in \\emptyset }\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ {} \\right\\}\\]<\/td>\n<\/tr>\n<tr>\n<td>h)<\/td>\n<td style=\"text-align: left;\">\\(64 = {x^2}\\)<\/td>\n<td style=\"text-align: left;\">\\[\\begin{array}{*{20}{l}}{64 = {x^2}}&amp; \\Leftrightarrow &amp;{{x^2} = 64}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 8}&amp; \\vee &amp;{x = 8}\\end{array}}\\end{array}\\]<\/td>\n<td style=\"text-align: left;\">\\[S = \\left\\{ { &#8211; 8,\\;8} \\right\\}\\]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0<\/p>\n<h6>Alternativa para a resolu\u00e7\u00e3o da equa\u00e7\u00e3o<\/h6>\n<p>Vamos resolver seguidamente algumas das equa\u00e7\u00f5es acima, utilizando o caso not\u00e1vel \\({a^2} &#8211; {b^2} = \\left( {a + b} \\right)\\left( {a &#8211; b} \\right)\\) e a Lei do Anulamento do Produto (\\(\\begin{array}{*{20}{l}}{A \\times B = 0}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{A = 0}&amp; \\vee &amp;{B = 0}\\end{array}}\\end{array}\\)).<\/p>\n<p>\u00a0<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{x^2} = 9}&amp; \\Leftrightarrow &amp;{{x^2} &#8211; 9 = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\left( {x + 3} \\right)\\left( {x &#8211; 3} \\right) = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x + 3 = 0}&amp; \\vee &amp;{x &#8211; 3 = 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 3}&amp; \\vee &amp;{x = 3}\\end{array}}\\end{array}\\]<\/p>\n<p>\u00a0<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{x^2} &#8211; 4 = 0}&amp; \\Leftrightarrow &amp;{\\left( {x + 2} \\right)\\left( {x &#8211; 2} \\right) = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x + 2 = 0}&amp; \\vee &amp;{x &#8211; 2 = 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 2}&amp; \\vee &amp;{x = 2}\\end{array}}\\end{array}\\]<\/p>\n<p>\u00a0<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{4{x^2} &#8211; 25 = 0}&amp; \\Leftrightarrow &amp;{\\left( {2x + 5} \\right)\\left( {2x &#8211; 5} \\right) = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{2x + 5 = 0}&amp; \\vee &amp;{2x &#8211; 5 = 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\frac{5}{2}}&amp; \\vee &amp;{x = \\frac{5}{2}}\\end{array}}\\end{array}\\]<\/p>\n<p>\u00a0<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{9{x^2} = 16}&amp; \\Leftrightarrow &amp;{9{x^2} &#8211; 16 = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\left( {3x + 4} \\right)\\left( {3x &#8211; 4} \\right) = 0}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{3x + 4 = 0}&amp; \\vee &amp;{3x &#8211; 4 = 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; \\frac{4}{3}}&amp; \\vee &amp;{x = \\frac{4}{3}}\\end{array}}\\end{array}\\]<\/p>\n<p>H\u00e1 vantagem em utilizar esta alternativa?<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_24625' onClick='GTTabs_show(0,24625)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Qual \u00e9 o conjunto-solu\u00e7\u00e3o de cada uma das seguintes equa\u00e7\u00f5es? a) \\({x^2} = 9\\) b) \\({x^2} &#8211; 4 = 0\\) c) \\(4{x^2} &#8211; 25 = 0\\) d) \\(9{x^2} = 16\\) e)&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19188,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,700],"tags":[424,706,198,705],"series":[],"class_list":["post-24625","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-monomios-e-polinomios","tag-8-o-ano","tag-equacao-incompleta-do-2-o-grau","tag-lei-do-anulamento-do-produto","tag-polinomios"],"views":121,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat74.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/24625","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=24625"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/24625\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19188"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=24625"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=24625"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=24625"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=24625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}