{"id":8596,"date":"2012-04-19T22:14:01","date_gmt":"2012-04-19T21:14:01","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8596"},"modified":"2022-01-06T22:17:49","modified_gmt":"2022-01-06T22:17:49","slug":"qual-deve-ser-o-valor","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8596","title":{"rendered":"Qual deve ser o valor?"},"content":{"rendered":"<p><ul id='GTTabs_ul_8596' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8596' class='GTTabs_curr'><a  id=\"8596_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8596' ><a  id=\"8596_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_8596'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Qual deve ser o valor de:<\/p>\n<ol>\n<li>$m$, para que a equa\u00e7\u00e3o $2{x^2} &#8211; 3mx + 2 = 0$ possua apenas uma raiz?<\/li>\n<li>$n$, para que a equa\u00e7\u00e3o ${x^2} &#8211; 6x + n &#8211; 4 = 0$ possua ra\u00edzes reais?<\/li>\n<li>$p$, para que a equa\u00e7\u00e3o $\\left( {2p + 1} \\right){x^2} &#8211; 3x + 1 = 0$ n\u00e3o possua ra\u00edzes reais?<\/li>\n<li>$r$, para que a equa\u00e7\u00e3o ${x^2} &#8211; 5x &#8211; r &#8211; 1 = 0$ tenha duas ra\u00edzes reais diferentes?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8596' onClick='GTTabs_show(1,8596)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8596'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Qual deve ser o valor de:<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote><p>$m$, para que a equa\u00e7\u00e3o $2{x^2} &#8211; 3mx + 2 = 0$ possua apenas uma raiz?<\/p><\/blockquote>\n<\/li>\n<li>\n<blockquote><p>$n$, para que a equa\u00e7\u00e3o ${x^2} &#8211; 6x + n &#8211; 4 = 0$ possua ra\u00edzes reais?<\/p><\/blockquote>\n<\/li>\n<li>\n<blockquote><p>$p$, para que a equa\u00e7\u00e3o $\\left( {2p + 1} \\right){x^2} &#8211; 3x + 1 = 0$ n\u00e3o possua ra\u00edzes reais?<\/p><\/blockquote>\n<\/li>\n<li>\n<blockquote><p>$r$, para que a equa\u00e7\u00e3o ${x^2} &#8211; 5x &#8211; r &#8211; 1 = 0$ tenha duas ra\u00edzes reais diferentes?<\/p><\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00ad<\/p>\n<ol>\n<li>$$2{x^2} &#8211; 3mx + 2 = 0$$<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n\\Delta &amp; = &amp;{{{\\left( { &#8211; 3m} \\right)}^2} &#8211; 4 \\times 2 \\times 2} \\\\<br \/>\n{}&amp; = &amp;{9{m^2} &#8211; 16}<br \/>\n\\end{array}$$<br \/>\nPara que a equa\u00e7\u00e3o possua apenas uma raiz ter\u00e1 de ser $\\Delta\u00a0 = 0$.<br \/>\nLogo, vem: $$\\begin{array}{*{20}{l}}<br \/>\n{\\Delta\u00a0 = 0}&amp; \\Leftrightarrow &amp;{9{m^2} &#8211; 16 = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{m^2} = \\frac{{16}}{9}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{m =\u00a0 &#8211; \\frac{4}{3}}&amp; \\vee &amp;{m = \\frac{4}{3}}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>$${x^2} &#8211; 6x + n &#8211; 4 = 0$$<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n\\Delta &amp; = &amp;{{{( &#8211; 6)}^2} &#8211; 4 \\times 1 \\times \\left( {n &#8211; 4} \\right)} \\\\<br \/>\n{}&amp; = &amp;{52 &#8211; 4n}<br \/>\n\\end{array}$$<br \/>\nPara que a equa\u00e7\u00e3o possua ra\u00edzes reais ter\u00e1 de ser $\\Delta\u00a0 \\geqslant 0$.<br \/>\nLogo, vem: $$\\begin{array}{*{20}{l}}<br \/>\n{\\Delta\u00a0 \\geqslant 0}&amp; \\Leftrightarrow &amp;{52 &#8211; 4n \\geqslant 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{4n \\leqslant 52} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n \\leqslant 13}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>$$\\left( {2p + 1} \\right){x^2} &#8211; 3x + 1 = 0$$<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n\\Delta &amp; = &amp;{{{( &#8211; 3)}^2} &#8211; 4 \\times \\left( {2p + 1} \\right) \\times 1} \\\\<br \/>\n{}&amp; = &amp;{5 &#8211; 8p}<br \/>\n\\end{array}$$<br \/>\nPara que a equa\u00e7\u00e3o n\u00e3o possua ra\u00edzes reais ter\u00e1 de ser $\\Delta\u00a0 &lt; 0$.<br \/>\nLogo, vem: $$\\begin{array}{*{20}{l}}<br \/>\n{\\Delta\u00a0 &lt; 0}&amp; \\Leftrightarrow &amp;{5 &#8211; 8p &lt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{8p &gt; 5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{p &gt; \\frac{5}{8}}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>$${x^2} &#8211; 5x &#8211; r &#8211; 1 = 0$$<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n\\Delta &amp; = &amp;{{{( &#8211; 5)}^2} &#8211; 4 \\times 1 \\times \\left( { &#8211; r &#8211; 1} \\right)} \\\\<br \/>\n{}&amp; = &amp;{29 + 4r}<br \/>\n\\end{array}$$<br \/>\nPara que a equa\u00e7\u00e3o tenha duas ra\u00edzes reais diferentes ter\u00e1 de ser $\\Delta\u00a0 &gt; 0$.<br \/>\nLogo, vem: $$\\begin{array}{*{20}{l}}<br \/>\n{\\Delta\u00a0 &gt; 0}&amp; \\Leftrightarrow &amp;{29 + 4r &gt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{r &gt;\u00a0 &#8211; \\frac{{29}}{4}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8596' onClick='GTTabs_show(0,8596)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Qual deve ser o valor de: $m$, para que a equa\u00e7\u00e3o $2{x^2} &#8211; 3mx + 2 = 0$ possua apenas uma raiz? $n$, para que a equa\u00e7\u00e3o ${x^2} &#8211; 6x +&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14114,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,303],"tags":[426,305,306],"series":[],"class_list":["post-8596","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-equacoes-do-2-o-grau","tag-9-o-ano","tag-binomio-discriminante","tag-equacao-do-2-o-grau"],"views":3524,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8596","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=8596"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8596\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8596"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8596"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8596"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8596"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}