{"id":81790,"date":"2017-08-16T20:21:48","date_gmt":"2017-08-16T23:21:48","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=81790"},"modified":"2023-04-08T02:18:30","modified_gmt":"2023-04-08T05:18:30","slug":"desafio-uma-equacao-curiosa","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/desafio-uma-equacao-curiosa\/","title":{"rendered":".Desafio: Uma equa\u00e7\u00e3o curiosa"},"content":{"rendered":"

\n

\n

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

\nDetermine [tex]A^2[\/tex], sabendo-se que [tex]A[\/tex] \u00e9 a soma dos valores absolutos de todas as ra\u00edzes da seguinte equa\u00e7\u00e3o: <\/p>\n

\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}}[\/tex]. <\/p>\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o 1 <\/p>\n

\nObserve, inicialmente, a seguinte sequ\u00eancia de igualdades equivalentes:<\/p>\n

\n[tex]\\quad \\begin{align*} &x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow x – \\sqrt{19} = \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}} \\;\\Longleftrightarrow \\\\
\n& \\Longleftrightarrow \\dfrac{\\color{red}{x – \\sqrt{19}}}{\\color{blue}{91}} = \\dfrac{\\color{red}{1}}{\\color{blue}{\\boxed{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}}}} \\;\\Longleftrightarrow \\\\
\n\\,\\\\
\n&\\Longleftrightarrow \\dfrac{\\color{blue}{91}}{\\color{red}{x – \\sqrt{19}}} =\\dfrac{\\color{blue}{ \\boxed{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}}}}{\\color{red}{1}}\\;\\Longleftrightarrow\\\\
\n\\,\\\\
\n&\\Longleftrightarrow \\dfrac{91}{x – \\sqrt{19}} = {\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow \\dfrac{91}{x – \\sqrt{19}} – \\sqrt{19}=\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}} \\;\\Longleftrightarrow \\\\
\n&\\Longleftrightarrow \\dfrac{91- \\sqrt{19}\\left(x – \\sqrt{19} \\right)}{x – \\sqrt{19}} =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow \\dfrac{91- \\sqrt{19}x +19}{x – \\sqrt{19}} =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow \\dfrac{110- \\sqrt{19}x}{x – \\sqrt{19}} =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}. \\qquad \\qquad \\color{#800000}(i)\\end{align*}[\/tex]<\/p>\n

Mas, perceba que: <\/p>\n

\n[tex]\\qquad \\begin{align*}\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}& =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\dfrac{\\sqrt{19}x+91}{x}}}}}}}\\\\
\n& =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}}\\end{align*}[\/tex]<\/p>\n

logo, por [tex]\\color{#800000}(i)[\/tex], segue que<\/p>\n

\n[tex]\\quad \\dfrac{110- \\sqrt{19}x}{x – \\sqrt{19}} =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}}[\/tex]<\/p>\n

e temos, ent\u00e3o, uma nova sequ\u00eancia de igualdades equivalentes: <\/p>\n

\n[tex]\\quad \\begin{align*}&\\dfrac{110- \\sqrt{19}x}{x – \\sqrt{19}} =\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow \\dfrac{110- \\sqrt{19}x}{91\\left(x – \\sqrt{19}\\right)} =\\dfrac{1}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}} \\;\\Longleftrightarrow \\\\
\n&\\Longleftrightarrow \\dfrac{\\color{red}{110- \\sqrt{19}x}}{\\color{blue}{91x – 91\\sqrt{19}}}=\\dfrac{\\color{red}{1}}{\\color{blue}{\\boxed{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}}}}\\; \\Longleftrightarrow \\\\
\n\\;\\\\
\n&\\Longleftrightarrow \\dfrac{\\color{blue}{91x – 91\\sqrt{19}}}{\\color{red}{110- \\sqrt{19}x}}=\\dfrac{\\color{blue}{\\boxed{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}}}}{\\color{red}{1}} \\;\\Longleftrightarrow \\\\
\n\\,\\\\
\n&\\Longleftrightarrow \\dfrac{91x – 91\\sqrt{19}}{110- \\sqrt{19}x} ={\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}} \\;\\Longleftrightarrow \\\\
\n&\\Longleftrightarrow \\dfrac{91x – 91\\sqrt{19}}{110- \\sqrt{19}x} -\\sqrt{19}=\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}} \\;\\Longleftrightarrow\\\\
\n&\\Longleftrightarrow \\dfrac{91x – 91\\sqrt{19}-110\\sqrt{19}+ 19x}{110- \\sqrt{19}x}=\\dfrac{91}{{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}\\;\\Longleftrightarrow\\\\
\n &\\Longleftrightarrow \\dfrac{\\color{red}{110x- 201\\sqrt{19}}}{\\color{blue}{91\\times 110- 91 \\sqrt{19}x}}=\\dfrac{\\color{red}{1}}{{\\color{blue}{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}\\;\\Longleftrightarrow \\end{align*}[\/tex]
\n[tex]\\quad \\begin{align*}\\Longleftrightarrow \\dfrac{\\color{blue}{91\\times 110- 91 \\sqrt{19}x}}{\\color{red}{110x- 201\\sqrt{19}}}&=\\dfrac{{\\color{blue}{\\sqrt{19}+\\dfrac{91x}{{\\sqrt{19}x+91}}}}}{\\color{red}{1}}\\\\
\n&=\\sqrt{19}+\\dfrac{91x}{\\sqrt{19}x+91}\\\\
\n&=\\dfrac{\\sqrt{19}\\sqrt{19}x+91\\sqrt{19}+91x}{\\sqrt{19}x+91}.\\end{align*}[\/tex]<\/p>\n

Assim,
\n[tex] \\qquad \\dfrac{91\\times 110- 91 \\sqrt{19}x}{110x- 201\\sqrt{19}}=\\dfrac{110x+91\\sqrt{19}}{\\sqrt{19}x+91}[\/tex],
\ndonde, multiplicando em cruz, obtemos
\n[tex]\\quad \\bcancel{{\\color{#800000}91\\times 110 \\sqrt{19} \\, x}}-{\\color{#0080FF}{91\\times 19 x^2}}+91^2 \\times 110-{\\color{#800000}91^2\\sqrt{19} \\, x}\\\\
\n\\,\\\\
\n\\quad{\\color{#0080FF}{110^2 x^2}}+\\bcancel{{\\color{#800000}91\\times 110 \\sqrt{19} \\, x}}-{\\color{#800000}110\\times 201\\sqrt{19} \\, x}-201 \\times 91 \\times 19[\/tex]
\nou ainda
\n[tex]\\quad{\\color{#0080Ff}\\left(110^2+19\\times 91\\right)x^2}+{\\color{#800000}\\left(91^2 \\sqrt{19} -110 \\times 201 \\sqrt{19} \\right)x}\\\\
\n\\quad \\quad -\\left(91^2 \\times 110+201 \\times 91 \\times 10 \\right)=0. [\/tex]
\nDessa forma, estamos procurando as ra\u00edzes da equa\u00e7\u00e3o
\n[tex] \\quad 13829x^2-13829 \\sqrt{19} \\, x-1258439=0 \\, [\/tex]
\nque simplificada fica
\n[tex] \\quad \\boxed{x^2-\\sqrt{19} \\, x-91=0}[\/tex].
\nPara resolver a equa\u00e7\u00e3o [tex] \\, x^2-\\sqrt{19} \\, x-91=0[\/tex], observamos que [tex]\\Delta=19+364=383[\/tex] e, portanto [tex]x=\\dfrac{\\sqrt{19}\\pm \\sqrt{383}}{2}[\/tex].
\nLogo, a soma dos valores absolutos das ra\u00edzes da equa\u00e7\u00e3o \u00e9
\n[tex]\\quad A=\\dfrac{\\sqrt{19}+\\sqrt{383}}{2}+\\dfrac{\\sqrt{383}- \\sqrt{19}}{2}=\\sqrt{383}[\/tex].
\nFinalmente, [tex] \\fcolorbox{black}{#eee0e5}{$A^2=383$}[\/tex].<\/p>\n

\n

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

\n

\n

Solu\u00e7\u00e3o 2 <\/p>\n

\nObserve a sequ\u00eancia de igualdades equivalentes:<\/p>\n

\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{\\color{red}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}}}=\\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\color{red}\\dfrac{91+x\\sqrt{19}}{x}}}}}}}}}
\n\\\\[\/tex]
\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\color{blue}\\dfrac{91}{{\\dfrac{91+x\\sqrt{19}}{x}}}}}}}}}=\\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\color{blue}\\dfrac{91x}{91+x\\sqrt{19}}}}}}}}
\n\\\\[\/tex]
\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\color{green}\\sqrt{19}+\\dfrac{91x}{91+x\\sqrt{19}}}}}}}} = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\color{green}\\dfrac{91\\sqrt{19}+110x}{91+x\\sqrt{19}}}}}}}}
\n\\\\[\/tex]
\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\color{red}\\dfrac{91}{{\\dfrac{91\\sqrt{19}+110x}{91+x\\sqrt{19}}}}}}}}= \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\color{red}\\dfrac{91^2+91x\\sqrt{19}}{91\\sqrt{19}+110x}}}}}
\n\\\\[\/tex]
\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\color{blue}\\sqrt{19}+\\dfrac{91^2+91x\\sqrt{19}}{91\\sqrt{19}+110x}}}}}=\\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\color{blue}\\dfrac{110\\times91+201x\\sqrt{19}}{91\\sqrt{19}+110x}}}}}
\n\\\\ \\, \\\\[\/tex]
\n[tex] x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\color{green}\\dfrac{91}{{\\dfrac{110\\times91+201x\\sqrt{19}}{91\\sqrt{19}+110x}}}}}=\\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+{\\color{green}\\dfrac{91^2\\sqrt{19}+91\\times110x}{110\\times91+201x\\sqrt{19}}}}}
\n\\\\ \\, \\\\[\/tex]
\n[tex]x = \\sqrt{19} + \\dfrac{91}{{\\color{red}\\sqrt{19}+{\\dfrac{91^2\\sqrt{19}+91\\times110x}{110\\times91+201x\\sqrt{19}}}}}=\\sqrt{19} + \\dfrac{91}{{{\\color{red}\\dfrac{201\\times91\\sqrt{19}+13829x}{110\\times91+201x\\sqrt{19}}}}}
\n\\\\ \\, \\\\[\/tex]
\n[tex] x = \\sqrt{19} +{\\color{blue} \\dfrac{91}{{{\\dfrac{201\\times91\\sqrt{19}+13829x}{110\\times91+201x\\sqrt{19}}}}}}= \\sqrt{19} + \\color{blue}\\dfrac{91({110\\times91+201x\\sqrt{19}})}{{{201\\times91\\sqrt{19}+13829x}}}\\\\ \\, \\\\[\/tex]
\n[tex] x = \\sqrt{19} + \\dfrac{\\color{blue}{91(110\\times91+201x\\sqrt{19})}}{{{201\\times91\\sqrt{19}+13829x}}}=\\sqrt{19} + \\dfrac{\\color{blue}{110\\times91^2 +91 \\times 201x\\sqrt{19}}}{{{201\\times91\\sqrt{19}+13829x}}}\\\\ \\, \\\\[\/tex]
\n[tex] x = \\color{green}{\\sqrt{19} + \\dfrac{110\\times91^2 +91 \\times 201x\\sqrt{19}}{{{201\\times91\\sqrt{19}+13829x}}}}=\\color{green}{ \\dfrac{\\sqrt{19}\\times (201\\times 91 \\sqrt{19}+13829x)+ 110\\times91^2 +91 \\times 201x\\sqrt{19}}{{{201\\times91\\sqrt{19}+13829x}}}}\\\\ \\, \\\\[\/tex]
\n[tex] x = \\dfrac{\\sqrt{19}\\times (201\\times 91 \\sqrt{19}+13829x)+ 110\\times91^2 +91 \\times 201x\\sqrt{19}}{{{201\\times91\\sqrt{19}+13829x}}}\\\\ \\, \\\\[\/tex]
\n[tex] x = \\dfrac{201\\times91 \\times 19+13829 \\sqrt{19}x+ 91^2 \\times 110+ 91 \\times 201x\\sqrt{19}}{{{201\\times91\\sqrt{19}+13829x}}}\\\\[\/tex]
\n[tex] x(201\\times91\\sqrt{19}+13829x) = 201\\times91 \\times 19+13829 \\sqrt{19}x+ 91^2 \\times 110+ 91 \\times 201x\\sqrt{19}\\\\[\/tex]
\n[tex] \\bcancel{201\\times91\\sqrt{19} \\, x}+13829x^2 = 201\\times91 \\times 19+13829 \\sqrt{19}x+ 91^2 \\times 110+ \\bcancel{91 \\times 201x\\sqrt{19}}\\\\[\/tex]
\n[tex] 13829x^2 = 201\\times91 \\times 19+13829 \\sqrt{19}x+ 91^2 \\times 110[\/tex]
\n[tex] 13829x^2 -13829 \\sqrt{19}x = 201\\times91 \\times 19+ 91^2 \\times 110[\/tex]
\n[tex] 13829x^2 -13829 \\sqrt{19}x = 1258439[\/tex]
\n[tex] 13829x^2 -13829 \\sqrt{19}x – 1258439=0[\/tex]
\n[tex] x^2 -\\sqrt{19}x – 91=0[\/tex]<\/p>\n

Dessa forma, vemos que a equa\u00e7\u00e3o original \u00e9 equivalente \u00e0 equa\u00e7\u00e3o do segundo grau [tex] \\boxed{x^2 -\\sqrt{19}x – 91=0}[\/tex], que \u00e9 muito simples de ser resolvida.
\nPara resolv\u00ea-la, basta observamos que [tex]\\Delta=19+364=383[\/tex]; portanto [tex]x=\\dfrac{\\sqrt{19}\\pm \\sqrt{383}}{2}[\/tex].
\nAssim, a soma dos valores absolutos das ra\u00edzes da equa\u00e7\u00e3o \u00e9 [tex]A=\\dfrac{\\sqrt{19}+\\sqrt{383}}{2}+\\dfrac{\\sqrt{383}- \\sqrt{19}}{2}=\\sqrt{383}[\/tex].
\nMais uma vez, [tex] \\fcolorbox{black}{#eee0e5}{$A^2=383$}[\/tex].<\/p>\n

\n

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

\n

\n\"\"<\/p>\n

Uma solu\u00e7\u00e3o tirada da cartola<\/p>\n

\n

Pessoas que estejam acostumadas a resolver problemas ol\u00edmpicos conhecem algumas ferramentas de Matem\u00e1tica elementar que aparecem quando se estuda um tema conhecido como “Teoria das Fra\u00e7\u00f5es Cont\u00ednuas<\/em><\/strong>” (ou continuadas). Embora este nosso problema n\u00e3o seja exatamente o que poder\u00edamos definir como um problema que envolva fra\u00e7\u00f5es cont\u00ednuas , pessoas que j\u00e1 resolveram muitos problemas desse tipo iniciariam a solu\u00e7\u00e3o do problema em quest\u00e3o impondo que a equa\u00e7\u00e3o
\n[tex]\\qquad x = \\sqrt{19} + \\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{{\\sqrt{19}+\\dfrac{91}{x}}}}}}}}}[\/tex][tex]\\qquad \\qquad {\\color{#800000}(I)}[\/tex]
\n\u00e9 equivalente \u00e0 equa\u00e7\u00e3o <\/p>\n

\n

[tex]\\qquad x = \\sqrt{19} + \\dfrac{91}{x}[\/tex], [tex]\\qquad \\qquad {\\color{#800000}(II)}[\/tex]
\no que de fato \u00e9 verdade.<\/p>\n

\n

Como na Matem\u00e1tica nossas afirma\u00e7\u00f5es carecem de justificativas, dependendo do contexto no qual um problema \u00e9 resolvido, as afirma\u00e7\u00f5es que utilizamos precisariam ser comprovadas.<\/p>\n