18-Resolvendo Equa\303\247\303\265es com Ra\303\255zes Quadradas Solucionar equa\303\247\303\265es com ra\303\255zes quadradas requer alguns passos adicionais, assim como a comprova\303\247\303\243o dos resultados.

restart; with(plots):

Warning, the name changecoords has been redefined

AdicioneAmbosLados := proc( expr, a ) lhs(expr) + a = rhs( expr) + a; end proc: MultipliqueAmbosLados := proc( expr, a ) simplify(lhs(expr) * a) = simplify( rhs( expr) * a) ; end proc: SimplifiqueAmbosLados := proc( expr ) simplify( expand( lhs(expr) )) = simplify( expand( rhs(expr) )); end proc: FatoreAmbosLados := proc( expr ) factor( lhs(expr) ) = factor( rhs(expr) ); end proc: EleveQuadradoAmobosLados := proc( expr ) simplify( lhs(expr)^2 ) = simplify( rhs(expr)^2 ); end proc:

Este \303\251 um tipo de problema bem direto. Problema : Encontre x : 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. Eq := sqrt( x + 3) = x - 2;

NiM+JSNFcUcvKiQpLCYlInhHIiIiIiIkIiIiIyIiIiIiIyIiIiwmJSJ4RyIiIiIiIyEiIg==

Passo 1. Observe que a raiz quadrada j\303\241 est\303\241 isolada, ent\303\243o, eleve os dois lados ao quadrado : EleveQuadradoAmobosLados( %);

NiMvLCYlInhHIiIiIiIkIiIiKiQpLCYlInhHIiIiIiIjISIiIiIjIiIi

Passo 2. Resolva e simplifique SimplifiqueAmbosLados( %);

NiMvLCYlInhHIiIiIiIkIiIiLCgqJCklInhHIiIjIiIiIiIiKiYiIiUiIiIlInhHIiIiISIiIiIlIiIi

Passo 3. Coloque tudo em um s\303\263 lado AdicioneAmbosLados( %, -lhs(%) );

NiMvIiIhLCgqJCklInhHIiIjIiIiIiIiKiYiIiYiIiIlInhHIiIiISIiIiIiIiIi

Passo 4. Resolva a equa\303\247\303\243o quadr\303\241tica. soln := solve(%, x);

NiM+JSVzb2xuRzYkLCYjIiImIiIjIiIiKiYiIiMhIiIpIiNAIyIiIiIiIyIiIiEiIiwmIyIiJiIiIyIiIiomIiIjISIiKSIjQCMiIiIiIiMiIiIiIiI=

Passo 5. O que geralmente \303\251 esquecido - Verifique as respostas! subs( x = soln[1], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvKiQpLCYjIiM2IiIjIiIiKiYiIiMhIiIpIiNAIyIiIiIiIyIiIiEiIiMiIiIiIiMiIiIsJiMiIiIiIiMiIiIqJiIiIyEiIikiI0AjIiIiIiIjIiIiISIiNiMvLCYqJiIiIyEiIikiI0AjIiIiIiIjIiIiIiIiIyIiIiIiIyEiIiwmIyIiIiIiIyIiIiomIiIjISIiKSIjQCMiIiIiIiMiIiIhIiI=NiMlJUZBSUxH

subs( x = soln[2], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvKiQpLCYjIiM2IiIjIiIiKiYiIiMhIiIpIiNAIyIiIiIiIyIiIiIiIiMiIiIiIiMiIiIsJiMiIiIiIiMiIiIqJiIiIyEiIikiI0AjIiIiIiIjIiIiIiIiNiMvLCYjIiIiIiIjIiIiKiYiIiMhIiIpIiNAIyIiIiIiIyIiIiIiIiwmIyIiIiIiIyIiIiomIiIjISIiKSIjQCMiIiIiIiMiIiIiIiI=NiMlJXRydWVH

a primeira solu\303\247\303\243o n\303\243o est\303\241 certa, enquanto a segunda funciona. Portanto, a solu\303\247\303\243o para o problema \303\251: Resposta := soln[2];

NiM+JSlSZXNwb3N0YUcsJiMiIiYiIiMiIiIqJiIiIyEiIikiI0AjIiIiIiIjIiIiIiIi

Este \303\251 um outro problema bem direto. Problema : Encontre x : 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 Eq := 3*x - sqrt( 2*x + 1) = 4*x - 9;

NiM+JSNFcUcvLCYqJiIiJCIiIiUieEciIiIiIiIqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiIiIiIiMiIiIiIiMiIiIhIiIsJiomIiIlIiIiJSJ4RyIiIiIiIiIiKiEiIg==

Passo 1. Isole a Raiz Quadrada AdicioneAmbosLados( %, -3*x);

NiMvLCQqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiIiIiIiMiIiIiIiMiIiIhIiIsJiUieEciIiIiIiohIiI=

Passo 2. Eleve os dois lados ao quadrado EleveQuadradoAmobosLados( %);

NiMvLCYqJiIiIyIiIiUieEciIiIiIiIiIiIiIiIqJCksJiUieEciIiIiIiohIiIiIiMiIiI=

Passo 3. Resolva e simplifique SimplifiqueAmbosLados( %);

NiMvLCYqJiIiIyIiIiUieEciIiIiIiIiIiIiIiIsKCokKSUieEciIiMiIiIiIiIqJiIjPSIiIiUieEciIiIhIiIiIyIpIiIi

Passo 4. Coloque tudo em um s\303\263 lado AdicioneAmbosLados( %, -lhs(%) );

NiMvIiIhLCgqJCklInhHIiIjIiIiIiIiKiYiIz8iIiIlInhHIiIiISIiIiMhKSIiIg==

Passo 5. Solucione a equa\303\247\303\243o quadr\303\241tica soln := solve(%, x);

NiM+JSVzb2xuRzYkLCYiIzUiIiIqJiIiIyIiIikiIiYjIiIiIiIjIiIiISIiLCYiIzUiIiIqJiIiIyIiIikiIiYjIiIiIiIjIiIiIiIi

Passo 6. Verifique as respostas ! subs( x = soln[1], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvLCgiI0kiIiIqJiIiJyIiIikiIiYjIiIiIiIjIiIiISIiKiQpLCYiI0AiIiIqJiIiJSIiIikiIiYjIiIiIiIjIiIiISIiIyIiIiIiIyIiIiEiIiwmIiNKIiIiKiYiIikiIiIpIiImIyIiIiIiIyIiIiEiIg==NiMvLCYiI0oiIiIqJiIiKSIiIikiIiYjIiIiIiIjIiIiISIiLCYiI0oiIiIqJiIiKSIiIikiIiYjIiIiIiIjIiIiISIiNiMlJXRydWVH

subs( x = soln[2], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvLCgiI0kiIiIqJiIiJyIiIikiIiYjIiIiIiIjIiIiIiIiKiQpLCYiI0AiIiIqJiIiJSIiIikiIiYjIiIiIiIjIiIiIiIiIyIiIiIiIyIiIiEiIiwmIiNKIiIiKiYiIikiIiIpIiImIyIiIiIiIyIiIiIiIg==NiMvLCYiI0giIiIqJiIiJSIiIikiIiYjIiIiIiIjIiIiIiIiLCYiI0oiIiIqJiIiKSIiIikiIiYjIiIiIiIjIiIiIiIiNiMlJUZBSUxH

Nesse caso, a primeira solu\303\247\303\243o est\303\241 certa, enquanto a segunda n\303\243o. Portanto, a solu\303\247\303\243o para o problema \303\251: Resposta := soln[1];

NiM+JSlSZXNwb3N0YUcsJiIjNSIiIiomIiIjIiIiKSIiJiMiIiIiIiMiIiIhIiI=

A coisa fica mais complexa quando temos duas ra\303\255zes quadradas. Basicamente, temos de elevar duas vezes ao quadrado, mas tendo o cuidado de isolar as ra\303\255zes quadradas em um lado na segunda vez. Problema : Encontre x : LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNidGKy1GIzYmLUkmbXNxcnRHRiQ2Iy1GIzYnRistRiM2Ji1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGRC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTC8lKXN0cmV0Y2h5R0ZMLyUqc3ltbWV0cmljR0ZMLyUobGFyZ2VvcEdGTC8lLm1vdmFibGVsaW1pdHNHRkwvJSdhY2NlbnRHRkwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zlbi1GLDYlUSJ4RidGL0YyRkQtRkc2LVEiK0YnRkRGSkZNRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GX28tRkE2JFEiM0YnRkRGRC1GRzYtUSgmbWludXM7RidGREZKRk1GT0ZRRlNGVUZXRl5vRmBvLUY6NiMtRiM2J0YrRj5GZG9GYW9GREZELUZHNi1RIj1GJ0ZERkpGTUZPRlFGU0ZVRlcvRlpRLDAuMjc3Nzc3OGVtRicvRmduRl9wLUZBNiRRIjRGJ0ZERkRGK0ZE. Eq := sqrt( 2*x + 3) - sqrt( 2*x - 3) = 4;

NiM+JSNFcUcvLCYqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCIiIiMiIiIiIiMiIiIiIiIqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCEiIiMiIiIiIiMiIiIhIiIiIiU=

Passo 1. J\303\241 que a raiz quadrada j\303\241 est\303\241 isolada, eleve os dois lados ao quadrado. EleveQuadradoAmobosLados( %); SimplifiqueAmbosLados( %);

NiMvKiQpLCYqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCIiIiMiIiIiIiMiIiIiIiIqJCksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCEiIiMiIiIiIiMiIiIhIiIiIiMiIiIiIzs=NiMvLCYqJiIiJSIiIiUieEciIiIiIiIqKCIiIyIiIiksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCIiIiMiIiIiIiMiIiIpLCYqJiIiIyIiIiUieEciIiIiIiIiIiQhIiIjIiIiIiIjIiIiISIiIiM7

Passo 2. Coloque tudo em um s\303\263 lado AdicioneAmbosLados( %, -4*x );

NiMvLCQqKCIiIyIiIiksJiomIiIjIiIiJSJ4RyIiIiIiIiIiJCIiIiMiIiIiIiMiIiIpLCYqJiIiIyIiIiUieEciIiIiIiIiIiQhIiIjIiIiIiIjIiIiISIiLCYiIzsiIiIqJiIiJSIiIiUieEciIiIhIiI=

Passo 3. Agora, eleve os dois lados ao quadrado novamente e resolva EleveQuadradoAmobosLados( %); SimplifiqueAmbosLados( %);

NiMvLCYqJiIjOyIiIiklInhHIiIjIiIiIiIiIiNPISIiLCQqJiIjOyIiIiksJiIiJSEiIiUieEciIiIiIiMiIiIiIiI=NiMvLCYqJiIjOyIiIiklInhHIiIjIiIiIiIiIiNPISIiLCgiJGMjIiIiKiYiJEciIiIiJSJ4RyIiIiEiIiomIiM7IiIiKSUieEciIiMiIiIiIiI=

Passo 4. Coloque tudo em um s\303\263 lado e resolva AdicioneAmbosLados( %, -lhs(%) );

NiMvIiIhLCYiJCNIIiIiKiYiJEciIiIiJSJ4RyIiIiEiIg==

Passo 5. Solucione a equa\303\247\303\243o quadr\303\241tica soln := solve(%, x);

NiM+JSVzb2xuRyMiI3QiI0s=

Passo 6. Verifique as respostas! subs( x = soln, Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvLCYqKCIjOyEiIikiJEAiIyIiIiIiIyIiIikiIzsjIiIiIiIjIiIiIiIiKigiIzshIiIpIiNEIyIiIiIiIyIiIikiIzsjIiIiIiIjIiIiISIiIiIlNiMvIyIiJCIiIyIiJQ==NiMlJmZhbHNlRw==

A \303\272nica solu\303\247\303\243o est\303\241 errada, isso significa que n\303\243o h\303\241 solu\303\247\303\243o para o problema! Resposta := {};

NiM+JSlSZXNwb3N0YUc8Ig==

Como pode ser? Vamos examinar a equa\303\247\303\243o original mais detalhadamente. Se plotarmos os dois lados da equa\303\247\303\243o original, veremos que as duas n\303\243o se intersecionam. \303\211 por isso que n\303\243o tem solu\303\247\303\243o. plot( { lhs(Eq), rhs(Eq)}, x = 1.5..100);

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A coisa fica mais complexa quando temos duas ra\303\255zes quadradas. Basicamente, temos de elevar duas vezes ao quadrado, mas novamente tendo o cuidado de isolar as ra\303\255zes quadradas em um lado na segunda vez. Problema : Encontre x : 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. Eq := sqrt( 7 - x) + sqrt( 7 + x) = x;

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Passo 1. Dado que a raiz quadrada j\303\241 est\303\241 isolada, eleve os dois lados ao quadrado. EleveQuadradoAmobosLados( %); SimplifiqueAmbosLados( %);

NiMvKiQpLCYqJCksJiIiKCIiIiUieEchIiIjIiIiIiIjIiIiIiIiKiQpLCYiIigiIiIlInhHIiIiIyIiIiIiIyIiIiIiIiIiIyIiIiokKSUieEciIiMiIiI=NiMvLCYiIzkiIiIqKCIiIyIiIiksJiIiKCIiIiUieEchIiIjIiIiIiIjIiIiKSwmIiIoIiIiJSJ4RyIiIiMiIiIiIiMiIiIiIiIqJCklInhHIiIjIiIi

Passo 2. Coloque tudo em um s\303\263 lado AdicioneAmbosLados( %, -14 );

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Passo 3. Agora, eleve os dois lados ao quadrado novamente e resolva EleveQuadradoAmobosLados( %); SimplifiqueAmbosLados( %);

NiMvLCYiJCc+IiIiKiYiIiUiIiIpJSJ4RyIiIyIiIiEiIiokKSwmKiQpJSJ4RyIiIyIiIiIiIiIjOSEiIiIiIyIiIg==NiMvLCYiJCc+IiIiKiYiIiUiIiIpJSJ4RyIiIyIiIiEiIiwoKiQpJSJ4RyIiJSIiIiIiIiomIiNHIiIiKSUieEciIiMiIiIhIiIiJCc+IiIi

Passo 4. Coloque tudo em um s\303\263 lado e resolva AdicioneAmbosLados( %, -lhs(%) ); FatoreAmbosLados( %);

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Passo 5. Solucione a equa\303\247\303\243o quadr\303\241tica. Existem tr\303\252s respostas soln := solve(%, x);

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Step 6. Verifique as respostas subs( x = soln[1], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

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subs( x = soln[2], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

NiMvLCQqJiIiIyIiIikiIigjIiIiIiIjIiIiIiIiIiIhNiMvLCQqJiIiIyIiIikiIigjIiIiIiIjIiIiIiIiIiIhNiMlJmZhbHNlRw==

subs( x = soln[3], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

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subs( x = soln[4], Eq); SimplifiqueAmbosLados( %); testeq( lhs(%) = rhs(%));

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H\303\241 apenas uma solu\303\247\303\243o Resposta := soln[3];

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