{"id":11854,"date":"2014-04-29T22:47:02","date_gmt":"2014-04-29T21:47:02","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11854"},"modified":"2022-01-12T00:02:32","modified_gmt":"2022-01-12T00:02:32","slug":"resolva-em-mathbbr-as-seguintes-equacoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11854","title":{"rendered":"Resolva, em $\\mathbb{R}$, as seguintes equa\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_11854' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11854' class='GTTabs_curr'><a  id=\"11854_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11854' ><a  id=\"11854_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_11854'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Resolva, em $\\mathbb{R}$, as seguintes equa\u00e7\u00f5es:<\/p>\n<ol>\n<li>$x + \\sqrt {2x}\u00a0 = 0$<\/li>\n<li>$x + 3 &#8211; \\sqrt {2x &#8211; 6}\u00a0 = 0$<\/li>\n<li>$\\sqrt {1 &#8211; x}\u00a0 + \\sqrt {2x}\u00a0 = 0$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11854' onClick='GTTabs_show(1,11854)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11854'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>$x + \\sqrt {2x}\u00a0 = 0$\n<p>O dom\u00ednio da condi\u00e7\u00e3o \u00e9 $D = \\mathbb{R}_0^ + $.<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{x + \\sqrt {2x}\u00a0 = 0}&amp; \\Leftrightarrow &amp;{\\sqrt {2x}\u00a0 =\u00a0 &#8211; x} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{2x = {x^2}} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{x\\left( {x &#8211; 2} \\right) = 0} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{x = 0}&amp; \\vee &amp;{x = 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<br \/>\n<span style=\"text-decoration: underline;\">Verifica\u00e7\u00e3o<\/span>:<\/p>\n<p>${x = 0}$ \u00e9 solu\u00e7\u00e3o da equa\u00e7\u00e3o, pois $0 + \\sqrt {2 \\times 0}\u00a0 = 0 \\Leftrightarrow 0 = 0$ \u00e9 uma proposi\u00e7\u00e3o verdadeira.<\/p>\n<p>${x = 2}$ n\u00e3o \u00e9 solu\u00e7\u00e3o da equa\u00e7\u00e3o, pois $2 + \\sqrt {2 \\times 2}\u00a0 = 0 \\Leftrightarrow 4 = 0$\u00a0\u00e9 uma proposi\u00e7\u00e3o falsa.<\/p>\n<p>Portanto, o conjunto solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9 $S = \\left\\{ 0 \\right\\}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>$x + 3 &#8211; \\sqrt {2x &#8211; 6}\u00a0 = 0$\n<p>O dom\u00ednio da condi\u00e7\u00e3o \u00e9 $D = \\left[ {3, + \\infty } \\right[$.<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{x + 3 &#8211; \\sqrt {2x &#8211; 6}\u00a0 = 0}&amp; \\Leftrightarrow &amp;{\\sqrt {2x &#8211; 6}\u00a0 = x + 3} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{2x &#8211; 6 = {x^2} + 6x + 9} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{{x^2} + 4x + 15 = 0} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{x \\in \\emptyset }<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, o conjunto solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9\u00a0$S = \\left\\{ {} \\right\\}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>$\\sqrt {1 &#8211; x}\u00a0 + \\sqrt {2x}\u00a0 = 0$\n<p>O dom\u00ednio da condi\u00e7\u00e3o \u00e9 $D = \\left\\{ {x \\in \\mathbb{R}:1 &#8211; x \\geqslant 0 \\wedge 2x \\geqslant 0} \\right\\} = \\left[ {0,1} \\right]$.<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{\\sqrt {1 &#8211; x}\u00a0 + \\sqrt {2x}\u00a0 = 0}&amp; \\Leftrightarrow &amp;{\\sqrt {2x}\u00a0 =\u00a0 &#8211; \\sqrt {1 &#8211; x} } \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{2x = 0}&amp; \\wedge &amp;{1 &#8211; x = 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\emptyset }<br \/>\n\\end{array}\\]<br \/>\nPortanto, o conjunto solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9 $S = \\left\\{ {} \\right\\}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11854' onClick='GTTabs_show(0,11854)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Resolva, em $\\mathbb{R}$, as seguintes equa\u00e7\u00f5es: $x + \\sqrt {2x}\u00a0 = 0$ $x + 3 &#8211; \\sqrt {2x &#8211; 6}\u00a0 = 0$ $\\sqrt {1 &#8211; x}\u00a0 + \\sqrt {2x}\u00a0 = 0$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19178,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,157],"tags":[422,369],"series":[],"class_list":["post-11854","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-com-radicais","tag-11-o-ano","tag-equacoes-com-radicais"],"views":1541,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat69.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11854","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=11854"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11854\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19178"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11854"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11854"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11854"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11854"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}