{"id":144967,"date":"2020-01-31T14:08:50","date_gmt":"2020-01-31T16:08:50","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=144967"},"modified":"2023-06-10T07:54:14","modified_gmt":"2023-06-10T10:54:14","slug":"problemao-ali-babao-e-a-decima-setima-de-suas-40-equacoes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-ali-babao-e-a-decima-setima-de-suas-40-equacoes\/","title":{"rendered":".Problem\u00e3o: Ali Bab\u00e3o e a d\u00e9cima s\u00e9tima de suas 40 equa\u00e7\u00f5es"},"content":{"rendered":"

\n

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Problema
\n (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n

\n

\n

(Olimp\u00edada Cearense de Matem\u00e1tica<\/strong> 1986 \u2013 Ensino Fundamental) <\/span> Resolva a equa\u00e7\u00e3o [tex]\\boxed{\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3}\\,.[\/tex]<\/p>\n

\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o<\/p>\n

\n

\n

Fazendo [tex]x-9=y^3[\/tex], obtemos [tex]\\boxed{x=y^3+9}[\/tex]; substituindo essa express\u00e3o na equa\u00e7\u00e3o do enunciado, segue que:
\n[tex]\\qquad \\sqrt[3]{y^3+18}-y=3[\/tex]
\n[tex]\\qquad \\sqrt[3]{y^3+18}=y+3\\, .[\/tex]<\/p>\n

\n

Elevando ambos os membros dessa \u00faltima igualdade ao cubo, vem:
\n[tex]\\qquad \\left(\\sqrt[3]{y^3+18}\\right)^3=\\left(y+3\\right)^3[\/tex]
\n[tex]\\qquad y^3+18=y^3+9y^2+27y+27[\/tex]
\n[tex]\\qquad 9y^2+27y+9=0[\/tex]
\n[tex]\\qquad y^2+3y+1=0[\/tex]
\ne resolvendo essa equa\u00e7\u00e3o do segundo grau, obtemos [tex]y=\\dfrac{-3 \\pm \\sqrt{5}}{2}[\/tex]. <\/p>\n

\n

Assim, [tex]\\boxed{x=y^3+9=\\left(\\dfrac{-3 \\pm \\sqrt{5}}{2}\\right)^3+9}[\/tex], de onde obtemos as solu\u00e7\u00f5es:<\/p>\n

[tex]\\qquad \\qquad \\fcolorbox{black}{#eee0e5}{$x_1=\\left(\\dfrac{-3 – \\sqrt{5}}{2}\\right)^3+9$}\\qquad [\/tex] e [tex]\\qquad \\fcolorbox{black}{#eee0e5}{$x_2=\\left(\\dfrac{-3 + \\sqrt{5}}{2}\\right)^3+9$}\\, .[\/tex]<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n

\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) (Olimp\u00edada Cearense de Matem\u00e1tica 1986 \u2013 Ensino Fundamental) Resolva a equa\u00e7\u00e3o [tex]\\boxed{\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3}\\,.[\/tex] Solu\u00e7\u00e3o Fazendo [tex]x-9=y^3[\/tex], obtemos [tex]\\boxed{x=y^3+9}[\/tex]; substituindo essa express\u00e3o na equa\u00e7\u00e3o do enunciado, segue que: [tex]\\qquad \\sqrt[3]{y^3+18}-y=3[\/tex] [tex]\\qquad \\sqrt[3]{y^3+18}=y+3\\, .[\/tex] Elevando ambos os membros dessa \u00faltima igualdade ao cubo, vem: [tex]\\qquad \\left(\\sqrt[3]{y^3+18}\\right)^3=\\left(y+3\\right)^3[\/tex] [tex]\\qquad y^3+18=y^3+9y^2+27y+27[\/tex] … <\/p>\n

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-144967","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/144967","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=144967"}],"version-history":[{"count":13,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/144967\/revisions"}],"predecessor-version":[{"id":194142,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/144967\/revisions\/194142"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=144967"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}