{"id":94168,"date":"2018-01-16T00:34:49","date_gmt":"2018-01-16T02:34:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=94168"},"modified":"2023-04-15T04:50:48","modified_gmt":"2023-04-15T07:50:48","slug":"problemao-ali-babao-e-a-segunda-de-suas-40-equacoes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-ali-babao-e-a-segunda-de-suas-40-equacoes\/","title":{"rendered":".Problem\u00e3o: Ali Bab\u00e3o e a segunda de suas 40 equa\u00e7\u00f5es"},"content":{"rendered":"<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:20px;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\"\/>\n<p>Determinar todas as solu\u00e7\u00f5es reais da seguinte equa\u00e7\u00e3o:<\/p>\n<p style=\"margin:20px\">\n<p>[tex]\\qquad \\qquad \\dfrac{x^2}{x-1}+\\sqrt{x-1}+\\dfrac{\\sqrt{x-1}}{x^2}=\\dfrac{x-1}{x^2}+\\dfrac{1}{\\sqrt{x-1}}+\\dfrac{x^2}{\\sqrt{x-1}}.[\/tex]<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:35px;\">\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\"\/>\n<p>Fazendo [tex]\\boxed{x^2=a} \\, [\/tex] e [tex] \\, \\boxed{\\sqrt{x-1}=b}[\/tex], temos tamb\u00e9m que [tex] \\, \\boxed{b^2=x-1}[\/tex] e, portanto, podemos reescrever assim a equa\u00e7\u00e3o dada:<\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad \\dfrac{x^2}{x-1}+\\sqrt{x-1}+\\dfrac{\\sqrt{x-1}}{x^2}=\\dfrac{x-1}{x^2}+\\dfrac{1}{\\sqrt{x-1}}+\\dfrac{x^2}{\\sqrt{x-1}}[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad \\dfrac{a}{b^2}+b+\\dfrac{b}{a}=\\dfrac{b^2}{a}+\\dfrac{1}{b}+\\dfrac{a}{b} [\/tex]<\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad a^2(1-b)+a(b^3-b)+b^3-b^4=0.\\qquad \\textcolor{#800000}{(i)}[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p><strong>(1)<\/strong> Se [tex]1-b\\ne 0[\/tex], temos que a equa\u00e7\u00e3o [tex]\\textcolor{#800000}{(i)}[\/tex] \u00e9 uma equa\u00e7\u00e3o do segundo grau na vari\u00e1vel [tex]a[\/tex] e, neste caso, temos que a soma de suas ra\u00edzes \u00e9 [tex]b^2+b[\/tex] e o produto das mesmas \u00e9 [tex]b^3[\/tex].<br \/>\nDessa forma, duas poss\u00edveis ra\u00edzes  de [tex]\\textcolor{#800000}{(i)}[\/tex] seriam [tex]\\boxed{ a_1=b} \\,  [\/tex] e [tex] \\, \\boxed{a_2=b^2} \\,  [\/tex].<br \/>\n<strong>(1.1)<\/strong> Se [tex] a=b[\/tex], ter\u00edamos [tex]x^2=\\sqrt{x-1} \\, [\/tex], donde [tex] x^4=x-1 \\, [\/tex], ou ainda, [tex] x^4+1=x[\/tex].<br \/>\nMas observe que se [tex]x[\/tex] \u00e9 um n\u00famero real, ent\u00e3o ou [tex]|x|\\ge 1[\/tex] ou [tex]0\\le |x| \\lt 1[\/tex].<\/p>\n<ul>\n<li> De [tex]|x|\\ge 1[\/tex] conclu\u00edmos facilmente que [tex]x^4 \\ge x[\/tex] e, portanto, a igualdade [tex] x^4+1=x[\/tex] \u00e9 imposs\u00edvel no conjunto dos n\u00fameros reais.<\/li>\n<li>Se  [tex]|x|\\lt 1[\/tex], como [tex]x^4\\ge0[\/tex] para todo [tex]x\\in\\mathbb{R}[\/tex], ter\u00edamos<br \/>\n[tex]\\qquad x^4+1 \\ge 1\\gt x[\/tex]<br \/>\ne assim a igualdade [tex] x^4+1=x[\/tex] tamb\u00e9m seria imposs\u00edvel no conjunto dos n\u00fameros reais.<\/li>\n<\/ul>\n<p><strong>(1.2)<\/strong>  Se [tex]a=b^2[\/tex], ent\u00e3o [tex]x^2-x+1=0[\/tex] e essa equa\u00e7\u00e3o n\u00e3o possui ra\u00edzes reais.<\/ul>\n<p>Por <strong>(1.1)<\/strong> e <strong>(1.2)<\/strong> n\u00e3o podemos ter [tex]1-b\\ne 0[\/tex].<br \/>\n<strong>(2)<\/strong> Como, necessariamente, [tex]1-b= 0[\/tex], ent\u00e3o [tex]b=1[\/tex] e, portanto, [tex]\\sqrt{x-1}=1[\/tex], donde [tex]x=2.[\/tex]<br \/>\nPelo exposto, [tex]x=2[\/tex] \u00e9 a \u00fanica raiz real da equa\u00e7\u00e3o.<\/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","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 9\u00ba ano do E. F.) Determinar todas as solu\u00e7\u00f5es reais da seguinte equa\u00e7\u00e3o: [tex]\\qquad \\qquad \\dfrac{x^2}{x-1}+\\sqrt{x-1}+\\dfrac{\\sqrt{x-1}}{x^2}=\\dfrac{x-1}{x^2}+\\dfrac{1}{\\sqrt{x-1}}+\\dfrac{x^2}{\\sqrt{x-1}}.[\/tex] Solu\u00e7\u00e3o Fazendo [tex]\\boxed{x^2=a} \\, [\/tex] e [tex] \\, \\boxed{\\sqrt{x-1}=b}[\/tex], temos tamb\u00e9m que [tex] \\, \\boxed{b^2=x-1}[\/tex] e, portanto, podemos reescrever assim a equa\u00e7\u00e3o dada: [tex]\\qquad \\dfrac{x^2}{x-1}+\\sqrt{x-1}+\\dfrac{\\sqrt{x-1}}{x^2}=\\dfrac{x-1}{x^2}+\\dfrac{1}{\\sqrt{x-1}}+\\dfrac{x^2}{\\sqrt{x-1}}[\/tex] [tex]\\qquad \\dfrac{a}{b^2}+b+\\dfrac{b}{a}=\\dfrac{b^2}{a}+\\dfrac{1}{b}+\\dfrac{a}{b} [\/tex] [tex]\\qquad a^2(1-b)+a(b^3-b)+b^3-b^4=0.\\qquad \\textcolor{#800000}{(i)}[\/tex] (1) Se [tex]1-b\\ne &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-ali-babao-e-a-segunda-de-suas-40-equacoes\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-94168","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/94168","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=94168"}],"version-history":[{"count":7,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/94168\/revisions"}],"predecessor-version":[{"id":187142,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/94168\/revisions\/187142"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=94168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}