<Text-field style="Heading 1" layout="Heading 1">Motiva\303\247\303\243o</Text-field>

Fun\303\247\303\243o do Segundo Grau: Resolvendo Equa\303\247\303\265es Quadr\303\241ticas por Completamento de Quadrados

N\303\263s freq\303\274entemente criamos modelos que descreve algo como uma fun\303\247\303\243o de tempo. A fun\303\247\303\243o LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn descreve como alguma quantidade, como a temperatura, o custo, etc. refere-se a tempo t. Qualquer quest\303\243o da forma "Quando \303\251 que o modelo assume o valor c ?" exigiria resolver uma equa\303\247\303\243o da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUYsNiZRImNGJ0YvRjJGNUYyRkA=. Assim, a capacidade de resolver equa\303\247\303\265es \303\251 essencial na aplica\303\247\303\243o da matem\303\241tica para a an\303\241lise e design do mundo real

As mais simples equa\303\247\303\265es s\303\243o as equa\303\247\303\265es lineares, equa\303\247\303\265es da forma 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. As habilidades necess\303\241rias para resolver tais equa\303\247\303\265es s\303\243o aritm\303\251ticas: subtrair LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn de ambos os lados e depois dividir por LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn.

A mais complicada equa\303\247\303\243o polinomial \303\251 a quadr\303\241tica, uma equa\303\247\303\243o que pode ser posta na forma 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. Esta \303\251 a equa\303\247\303\243o quadr\303\241tica, cuja solu\303\247\303\243o pode, em casos raros, ser obtida atrav\303\251s de fatora\303\247\303\243o. Em geral, por\303\251m, tais equa\303\247\303\265es requerem o uso da f\303\263rmula quadr\303\241tica, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JJm1mcmFjR0YkNigtRiM2Ki1GOTYuUSomdW1pbnVzMDtGJ0YyRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUEZaLUYsNiZRImJGJ0YvRjJGNS1GOTYuUSJ+RidGMkY8Rj5GQEZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBGXW8tRjk2LlElJnBtO0YnRjJGPEY+RkBGQkZERkZGSEZKRllGZW4tRiM2Ji1GLDYjUSFGJy1JJm1zcXJ0R0YkNiMtRiM2Ky1JJW1zdXBHRiQ2JUZmbi1GIzYlLUkjbW5HRiQ2JVEiMkYnRjJGPEYyRjwvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY5Ni5RKCZtaW51cztGJ0YyRjxGPkZARkJGREZGRkhGSkZZRmVuLUZicDYlUSI0RidGMkY8RmluLUYsNiZRImFGJ0YvRjJGNUZpbi1GLDYmUSJjRidGL0YyRjVGMkY8RjJGPEZkb0YyRjwtRiM2J0ZhcEZpbkZecUYyRjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltyLyUpYmV2ZWxsZWRHRjRGMkY8, uma solu\303\247\303\243o expl\303\255cita obtido pelo processo de completar o quadrado.

Na aus\303\252ncia de resist\303\252ncia do ar, a altura de uma bola lan\303\247ada \303\251 parab\303\263lica, uma curva descrita por uma fun\303\247\303\243o quadr\303\241tica. Quest\303\265es sobre essa trajet\303\263ria, a sua altura em uma determinada dist\303\242ncia do ponto de lan\303\247amento, por exemplo, exigiria resolver uma equa\303\247\303\243o quadr\303\241tica. \303\211 por isso que tanta \303\252nfase \303\251 colocada sobre a fun\303\247\303\243o quadr\303\241tica, e a solu\303\247\303\243o de equa\303\247\303\265es quadr\303\241ticas, especialmente a solu\303\247\303\243o pela t\303\251cnica de completar o quadrado.

<Text-field style="Heading 1" layout="Heading 1">Resolvendo por Completamento de Quadrados</Text-field>

A maioria das equa\303\247\303\265es quadr\303\241ticas n\303\243o vem na forma de um quadrado, ou seja, na forma 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. Na verdade, a maioria das equa\303\247\303\265es quadr\303\241ticas vem na forma mais geral 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. Completando o quadrado d\303\241-lhe a forma 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. A partir daqui, \303\251 f\303\241cil de obter a solu\303\247\303\243o:

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

Etapas

Resolvendo uma equa\303\247\303\243o quadr\303\241tica completando o quadrado

(1)

Complete o quadrado se a equa\303\247\303\243o quadr\303\241tica j\303\241 n\303\243o tem essa forma.

(2)

Isolar o termo quadrado, adicionando a ambos os lados e, talvez, dividindo ambos os lados pelo coeficiente, se n\303\243o for 1.

(3)

Elevar cada lado da equa\303\247\303\243o para a pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JlEiMUYnLyUlc2l6ZUdRIzEwRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYvUSIvRidGL0YyRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZNLUYsNiZRIjJGJ0YvRjJGNUYvRjJGNQ==, isto \303\251, extrair a raiz quadrada de cada lado da equa\303\247\303\243o. Certifique-se de escrever tanto o positivo e o negativo da raiz.

(4)

Isolar o LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn no lado esquerdo, adicionando a quantidade adequada para cada lado da equa\303\247\303\243o. Deve haver duas solu\303\247\303\265es para x.

<Text-field style="Heading 1" layout="Heading 1">Demonstra\303\247\303\243o</Text-field>

Explorar as etapas da t\303\251cnica de completar o quadrado com esta demonstra\303\247\303\243o. Clique no bot\303\243o "Gere um Polin\303\264mio Quadratico" para gerar um novo polin\303\264mio quadr\303\241tico, e clique no "Execute esse Passo" para ver os resultados de cada etapa.

Gerar um Polinomial Quadr\303\241tico:

use DocumentTools in #Clear Boxes before bringing in the next question seq(SetProperty(i,value,NULL),i in [MathContainer1, MathContainer2, MathContainer3, MathContainer4, MathContainer5, MathContainer6, MathContainer7]); _myA, _myB, _myC := op(RandomTools:-Generate([integer(range = -10 .. 10), integer(range = -20 .. -1), integer(range = 1 .. 3)])); if _myA = 0 then _myA := _myB; _myB := 0; end if; _myEq := expand(_myC*(x+_myA)^2+_myB); Do(%MathContainer0=_myEq); _NextStep:=1; end use;

Equa\303\247\303\243o Original:

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Etapa 1:

Agrupar os termos da vari\303\241vel.

use DocumentTools in #Update section if _myEq<>NULL then Do(%MathContainer0=_myEq); _Temp := PolynomialTools[CoefficientList](collect(_myEq, x), x); if _NextStep=1 then CFIn:=coeff(_Temp,x,2); _TempCF := _Temp[3]; _Temp[2] := _Temp[2]/_TempCF; SetProperty(MathContainer1, value, MathML:-ExportPresentation(``*_TempCF*(x^2+_Temp[2]*x)+_Temp[1])); _NextStep:=2; end if; end if; end use;

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Equa\303\247\303\243o ap\303\263s Etapa 1:

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Etapa 2:

Para completar o quadrado, somar e subtrair o quadrado da metade do termo m\303\251dio.

use DocumentTools in #Update section if _NextStep=2 then _SqIn:=(_Temp[2]/2)^2; SetProperty(MathContainer2, value, MathML:-ExportPresentation(``*_TempCF*(x^2+_Temp[2]*x+_SqIn)+``*_Temp[1]-_SqIn*_TempCF)); _NextStep:=3; end if; end use;

Equa\303\247\303\243o ap\303\263s Etapa 2:

Etapa 3:

Escreva os tr\303\252s termos dentro dos par\303\252nteses como um quadrado perfeito e simplifique.

use DocumentTools in #Update section if _NextStep=3 then SetProperty(MathContainer3, value, MathML:-ExportPresentation(``*_TempCF*(x+_myA)^2+_Temp[1]-_SqIn*_TempCF)); _NextStep:=4; end if; end use;

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Equa\303\247\303\243o ap\303\263s Etapa 3:

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Etapa 4:

Igualar a equa\303\247\303\243o a zero.

use DocumentTools in #Update section if _NextStep=4 then SetProperty(MathContainer4, value, MathML:-ExportPresentation(``*_TempCF*(x+_myA)^2+_Temp[1]-_SqIn*_TempCF=0)); _NextStep:=5; end if; end use;

Equa\303\247\303\243o ap\303\263s Etapa 4:

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Etapa 5:

Isolar o termo quadrado

use DocumentTools in #Update section if _NextStep=5 then _Temp2:= [_myA,1]; SetProperty(MathContainer5, value, MathML:-ExportPresentation(``*(_Temp2[2]*x+_Temp2[1])^2= -(_Temp[1]-_SqIn*_TempCF)/_TempCF)); _NextStep:=6; end if; end use;

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Equa\303\247\303\243o ap\303\263s Etapa 5:

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Etapa 6:

Tome a raiz quadrada de cada lado da equa\303\247\303\243o.

use DocumentTools in #Update section if _NextStep=6 then SetProperty(MathContainer6, value, MathML:-ExportPresentation(``*(_Temp2[2]*x+_Temp2[1])= `±`*convert(sqrt(-(_Temp[1]-_SqIn*_TempCF)/_TempCF), '`local`'))); _NextStep:=7; end if; end use;

Equa\303\247\303\243o ap\303\263s Etapa 6:

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Etapa 7:

Isolar o termo x.

use DocumentTools in #Update section if _NextStep=7 then SetProperty(MathContainer7, value, MathML:-ExportPresentation(x = -_Temp2[1]*`±`*convert(sqrt(-(_Temp[1]-_SqIn*_TempCF)/_TempCF), '`local`'))); _NextStep:=8; end if; end use;

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Equa\303\247\303\243o ap\303\263s Etapa 7:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH

use DocumentTools in seq(SetProperty(i,value,NULL),i in [MathContainer1, MathContainer2, MathContainer3, MathContainer4, MathContainer5, MathContainer6, MathContainer7]); _NextStep:=1; end use;

use DocumentTools in seq(SetProperty(i,value,NULL),i in [MathContainer0, MathContainer1, MathContainer2, MathContainer3, MathContainer4, MathContainer5, MathContainer6, MathContainer7]); _myEq:=NULL; end use;

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