{"id":152206,"date":"2020-04-23T23:00:20","date_gmt":"2020-04-24T02:00:20","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=152206"},"modified":"2023-06-11T01:08:55","modified_gmt":"2023-06-11T04:08:55","slug":"desafio-valor-da-expressao","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/desafio-valor-da-expressao\/","title":{"rendered":".Desafio:  Valor de uma express\u00e3o"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:26px;\">\n<p class=\"tit\" style=\"line-height:20px; margin-bottom: 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><span style=\"font-size:12px;\"> (Extra\u00eddo do Material do <strong>PECI<\/strong>)<\/span> Sejam [tex]x_1, y_1, x_2, y_2[\/tex] n\u00fameros reais satisfazendo as equa\u00e7\u00f5es <\/p>\n<ul>\n<li>[tex]x_1^2+5x_2^2=10[\/tex], <\/li>\n<li>[tex]x_2y_1-x_1y_2=5[\/tex],<\/li>\n<li>[tex]x_1y_1+5x_2y_2=\\sqrt{105}[\/tex]. <\/li>\n<\/ul>\n<p><strong>Determinar o valor da express\u00e3o [tex]y_1^2+5y_2^2[\/tex].<\/strong><\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:33px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 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>Vamos iniciar calculando o produto [tex](x_1^2+5x_2^2)(y_1^2+5y_2^2)[\/tex]:<br \/>\n[tex]\\qquad (x_1^2+5x_2^2)(y_1^2+5y_2^2)=x_1^2y_1^2+5x_1^2y_2^2+5x_2^2y_1^2+25x_2^2y_2^2[\/tex].<br \/>\nAgora, vamos adicionar e subtrair a express\u00e3o [tex]10x_1y_1x_2y_2[\/tex] no segundo membro da igualdade anterior:<br \/>\n<span style=\"font-size:13.5px\">[tex]\\;\\;(x_1^2+5x_2^2)(y_1^2+5y_2^2)=x_1^2y_1^2+10x_1y_1x_2y_2-10x_1y_1x_2y_2+5x_1^2y_2^2+5x_2^2y_1^2+25x_2^2y_2^2[\/tex]<br \/>\n[tex]\\;\\;(x_1^2+5x_2^2)(y_1^2+5y_2^2)=(x_1^2y_1^2+10x_1y_1x_2y_2+25x_2^2y_2^2)+5(x_1^2y_2^2-2x_1y_1x_2y_2+x_2^2y_1^2).[\/tex]<\/span><br \/>\nObserve que h\u00e1 dois quadrados perfeitos no segundo membro da \u00faltima igualdade:<br \/>\n[tex]\\qquad (x_1^2+5x_2^2)(y_1^2+5y_2^2)=(x_1y_1+5x_2y_2)^2+5(x_2y_1-x_1y_2)^2[\/tex].<br \/>\nFinalmente, como sabemos que [tex]\\fcolorbox{red}{#ffffff}{$ x_1^2+5x_2^2=10$}[\/tex], [tex]\\fcolorbox{blue}{#ffffff}{$x_2y_1-x_1y_2=5$}[\/tex] e [tex]\\fcolorbox{#FF00FF}{#ffffff}{$x_1y_1+5x_2y_2=\\sqrt{105}$}[\/tex], temos<br \/>\n[tex]\\qquad \\textcolor{red}{10}(y_1^2+5y_2^2)=(\\textcolor{#FF00FF}{\\sqrt{105}})^2+\\textcolor{blue}{5}(5)^2[\/tex]<br \/>\ndonde<br \/>\n[tex]\\qquad 10(y_1^2+5y_2^2)=105+125=230[\/tex]<br \/>\n[tex]\\qquad \\fcolorbox{black}{#eee0e5}{$y_1^2+5y_2^2=23$}\\,.[\/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.) (Extra\u00eddo do Material do PECI) Sejam [tex]x_1, y_1, x_2, y_2[\/tex] n\u00fameros reais satisfazendo as equa\u00e7\u00f5es [tex]x_1^2+5x_2^2=10[\/tex], [tex]x_2y_1-x_1y_2=5[\/tex], [tex]x_1y_1+5x_2y_2=\\sqrt{105}[\/tex]. Determinar o valor da express\u00e3o [tex]y_1^2+5y_2^2[\/tex]. Solu\u00e7\u00e3o Vamos iniciar calculando o produto [tex](x_1^2+5x_2^2)(y_1^2+5y_2^2)[\/tex]: [tex]\\qquad (x_1^2+5x_2^2)(y_1^2+5y_2^2)=x_1^2y_1^2+5x_1^2y_2^2+5x_2^2y_1^2+25x_2^2y_2^2[\/tex]. Agora, vamos adicionar e subtrair a express\u00e3o [tex]10x_1y_1x_2y_2[\/tex] no segundo membro da &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/desafio-valor-da-expressao\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-152206","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/152206","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=152206"}],"version-history":[{"count":24,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/152206\/revisions"}],"predecessor-version":[{"id":194290,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/152206\/revisions\/194290"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=152206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}