{"id":103198,"date":"2018-05-17T17:54:16","date_gmt":"2018-05-17T20:54:16","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=103198"},"modified":"2023-04-20T21:53:39","modified_gmt":"2023-04-21T00:53:39","slug":"problemao-os-possiveis-valores-reais","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-os-possiveis-valores-reais\/","title":{"rendered":".Problem\u00e3o: Os poss\u00edveis valores reais"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nDetermine os poss\u00edveis valores reais de [tex]x[\/tex] que satisfazem a igualdade [tex]\\boxed{\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3}[\/tex].<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:32px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nAcompanhe passo a passo a resolu\u00e7\u00e3o.<\/p>\n<ul>\n<li>Elevando cada membro da equa\u00e7\u00e3o [tex]\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3[\/tex] ao cubo, temos:<br \/>\n[tex]\\qquad (\\sqrt[3]{x+9}-\\sqrt[3]{x-9})^3=3^3[\/tex]<br \/>\n[tex]\\qquad x+9-3(\\sqrt[3]{x+9})^2\\sqrt[3]{x-9}+3\\sqrt[3]{x+9}(\\sqrt[3]{x-9})^2-(x-9)=27\\quad \\textcolor{#800000}{(i)}[\/tex]<\/li>\n<li>Simplificando a igualdade [tex] \\textcolor{#800000}{(i)}[\/tex], temos:<br \/>\n[tex]\\qquad -3(\\sqrt[3]{x+9})^2\\sqrt[3]{x-9}+3\\sqrt[3]{x+9}(\\sqrt[3]{x-9})^2=27-18=9\\quad \\textcolor{#800000}{(ii)}[\/tex]<\/li>\n<li>Dividindo a equa\u00e7\u00e3o [tex] \\textcolor{#800000}{(ii)}[\/tex] por 3, obtemos:<br \/>\n[tex]\\qquad -(\\sqrt[3]{x+9})^2\\sqrt[3]{x-9}+\\sqrt[3]{x+9}(\\sqrt[3]{x-9})^2=3\\quad \\textcolor{#800000}{(iii)}[\/tex]<\/li>\n<li>Colocando em evid\u00eancia os fatores [tex]\\sqrt[3]{x+9}[\/tex]  e [tex]\\sqrt[3]{x-9}[\/tex] em [tex] \\textcolor{#800000}{(iii)} \\, [\/tex], podemos afirmar que:<br \/>\n[tex]\\qquad \\sqrt[3]{x+9}\\sqrt[3]{x-9}(-\\sqrt[3]{x+9}+\\sqrt[3]{x-9})=3\\quad \\textcolor{#800000}{(iv)}[\/tex]<\/li>\n<li>E usando em [tex] \\textcolor{#800000}{(iv)}[\/tex] o fato de que [tex]\\boxed{-\\sqrt[3]{x+9}+\\sqrt[3]{x-9}=-3}[\/tex], tem-se que:<br \/>\n[tex]\\qquad \\sqrt[3]{x+9}\\sqrt[3]{x-9}(-3)=3 \\quad \\textcolor{#800000}{(v)}[\/tex]<\/li>\n<li>Logo, segue de [tex] \\textcolor{#800000}{(v)}[\/tex] que:<br \/>\n[tex]\\qquad \\sqrt[3]{x+9}\\sqrt[3]{x-9}=-1[\/tex]<br \/>\n[tex]\\qquad \\sqrt[3]{(x+9)\\cdot (x-9)}=-1[\/tex]<br \/>\n[tex]\\qquad \\sqrt[3]{x^2-81}=-1[\/tex]<br \/>\n[tex]\\qquad \\left(\\sqrt[3]{x^2-81}\\right)^3=(-1)^3[\/tex]<br \/>\n[tex]\\qquad x^2-81=-1[\/tex]<br \/>\n[tex]\\qquad x^2=80[\/tex]<br \/>\n[tex]\\qquad x=\\pm 4\\sqrt{5}[\/tex].<\/li>\n<\/ul>\n<p>Portanto, [tex] \\, \\fcolorbox{black}{#eee0e5}{$x=4\\sqrt{5}$} \\, [\/tex] ou [tex] \\, \\fcolorbox{black}{#eee0e5}{$x=-4\\sqrt{5}$} \\, .[\/tex]<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 9\u00ba ano do E. F.) Determine os poss\u00edveis valores reais de [tex]x[\/tex] que satisfazem a igualdade [tex]\\boxed{\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3}[\/tex]. Solu\u00e7\u00e3o Acompanhe passo a passo a resolu\u00e7\u00e3o. Elevando cada membro da equa\u00e7\u00e3o [tex]\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3[\/tex] ao cubo, temos: [tex]\\qquad (\\sqrt[3]{x+9}-\\sqrt[3]{x-9})^3=3^3[\/tex] [tex]\\qquad x+9-3(\\sqrt[3]{x+9})^2\\sqrt[3]{x-9}+3\\sqrt[3]{x+9}(\\sqrt[3]{x-9})^2-(x-9)=27\\quad \\textcolor{#800000}{(i)}[\/tex] Simplificando a igualdade [tex] \\textcolor{#800000}{(i)}[\/tex], temos: [tex]\\qquad -3(\\sqrt[3]{x+9})^2\\sqrt[3]{x-9}+3\\sqrt[3]{x+9}(\\sqrt[3]{x-9})^2=27-18=9\\quad \\textcolor{#800000}{(ii)}[\/tex] Dividindo a equa\u00e7\u00e3o &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-os-possiveis-valores-reais\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":620,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-103198","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/103198","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/620"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=103198"}],"version-history":[{"count":38,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/103198\/revisions"}],"predecessor-version":[{"id":187647,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/103198\/revisions\/187647"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=103198"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}