15-F\303\263rmula Quadr\303\241tica & DiscriminanteEstudaremos a deriva\303\247\303\243o e uso da f\303\263rmula quadr\303\241tica e do discriminante.

restart; Quadr\303\241tica := proc( expr ) local a, b, c, soln, k; a := coeff(expr,x,2); b := coeff(expr,x,1); c := coeff(expr,x,0); printf("\134t\134t\134t\134t Equa\303\247\303\243o Quadr\303\241tica : \134t%A = 0\134n",expr); printf("\134t\134t\134t\134t Coeficientes :\134t\134t\134t\134t\134t\134t\134t\134ta=%d\134t\134t\134t b=%d\134t\134t\134t c=%d\134n", a,b,c); soln := [solve( expr = 0, x)]; if(nops(soln) = 1) then printf("\134t\134t\134t\134t Solu\303\247\303\243o : %A\134n\134n",soln[1]); soln[1]; else printf("\134t\134t\134t\134t Solut\303\247\303\243o :\134t\134t\134t\134t\134t\134t\134t\134t\134t\134t\134t%A\134t\134t%A\134n\134n", soln[1],soln[2]); soln[1],soln[2]; fi; end proc: Discriminante := proc( expr ) local a, b, c, soln, k, D; a := coeff(expr,x,2); b := coeff(expr,x,1); c := coeff(expr,x,0); printf("\134t\134t\134t\134t Equa\303\247\303\243o Quadr\303\241tica : \134t%A = 0\134n",expr); printf("\134t\134t\134t\134t Coeficiente :\134t\134t\134t\134t\134t\134t\134t\134ta=%d\134t\134t\134t b=%d\134t\134t\134t c=%d\134n", a,b,c); D := b*b - 4*a*c; printf("\134t\134t\134t\134t Discriminante =\134t\134t\134t\134t\134t\134t\134t\134t%d\134n",D); if( D > 0) then printf("\134t\134t\134t\134t Existem duas ra\303\255zes reais e distintas.\134n\134n"); else if( D < 0) then printf("\134t\134t\134t\134t Existem duas ra\303\255zes complexas e distintas.\134n\134n"); else printf("\134t\134t\134t\134t Existe uma raiz real. Essa express\303\243o quadr\303\241tica \303\251 um quadrado perfeito.\134n\134n"); fi;fi; if( D = floor(sqrt(D))^2) then printf("\134t\134t\134t\134t Essa express\303\243o \303\251 fator\303\241vel. \134n\134n"); print( factor(expr)); else printf("\134t\134t\134t\134t Essa express\303\243o N\303\203O \303\251 fator\303\241vel. \134n\134n"); fi; print( solve( expr, x) ); end proc: completandoQuadradocomplicado := proc(EQ) #ONLY for case when leading coef\037\302\255 1 <<DUPLICATE of other worksheet>> local EQ2,EQ3,A,B,C,b, lt, sg, subEQ, rEQ,g ; use RealDomain in print(`Equa\303\247\303\243o Original:`); print(EQ); subEQ := sort(op(1,EQ)-op(2,EQ), x); A := coeff(subEQ,x,2); B := coeff(subEQ,x,1); C := coeff(subEQ,x,0); g := A; if( B <> 0 ) then g := gcd( g, B); fi; if( C <> 0 ) then g := gcd( g, C); fi; EQ3 := (subEQ - C)/g = - C/g; lt := coeff((subEQ/g),x,2); if(lt=1) then CompleteSteps(EQ3); else print(`\134n1. Fatore cada um dos termos comuns, e `, `coloque os termos x a esquerda, e os termos num\303\251ricos a direita:`); print(EQ3); print(` `); A := A/g; sg := A/abs(A); if( sg = -1) then EQ3 := -lhs(EQ3) = -rhs(EQ3); print(`(Se o coeficiente de x^2 \303\251 negativo,`, ` multiplique ambos os lados por -1)`); print(EQ3); print(` `); fi; lt := abs(A); subEQ := lhs(EQ3)/lt; rEQ := rhs(EQ3); print(`Fatore o coeficiente principal a partir do lado esquerdo`); print(cat(lt,`(`,subEQ,`) = `, rEQ)); b := coeff( subEQ, x); print(`\134n2. Pegue a metade do coeficiente x`,b, ` e eleve ao quadrado para obter `, (b/2)^2); print(`..Ent\303\243o multiplique isto pelo fatores principais `,lt, ` e adicione aos dois lados:`); subEQ := subEQ + (b/2)^2; print(cat( lt,`(`,subEQ,`) = `,rEQ,` + `,lt,`*`,(b/2)^2, `)` )); rEQ := rEQ + lt*(b/2)^2; EQ2 := lt*subEQ = rEQ; print(` `); print(` `); print(`\134n3. Tome o quadrado de cada lado (com +/-)`); EQ2 := factor(lhs(EQ2)) = rhs(EQ2); print(EQ2); EQ3 := sqrt(lhs(%)) = sqrt(rhs(%)); lt:= op(1, lhs(EQ3)); rEQ := rhs(EQ3); subEQ := op(1, op(2, lhs(EQ2))); print(` `); print(`\134n4. Agora resolva as duas equa\303\247\303\265es lineares:`); print(cat( lt,`(`, subEQ ,`) = `, rEQ )); print(cat( lt,`(`, convert(subEQ, string) ,`) = `, -rEQ )); print(` `); print( lt*subEQ = rEQ); print( lt*subEQ = -rEQ); print(` `); print(x = {solve(EQ2, x)}); fi; end use; end proc:

Voc\303\252 deve lembrar que podemos resolver qualquer equa\303\247\303\243o quadr\303\241tica por completando o quadrado. CompleteStepsMore( 3*x^2 + 30*x + 1 = 0);

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Cada um desses prolemas \303\251 um tanto longo e teremos de fazer os mesmos passos a cada vez. Ent\303\243o existe uma maneira de ir direto ao ponto e obter o resultado por esse m\303\251todo mais r\303\241pido? E se fizermos por um polin\303\264mio quadr\303\241tico qualquer com letras em vez de n\303\272meros? CompleteStepsMore( a*x^2 + b*x + c = 0);

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Consideremos a f\303\263rmula acima. Essa \303\251 a f\303\263rmula quadr\303\241tica. Ela geralmente \303\251 escrita de uma forma diferente em textos, como uma \303\272nica fra\303\247\303\243o com um 2a no denomidador e um sinal de "mais ou menos" em frente da raiz quadrada para servir para os dois casos ao mesmo tempo. Nos d\303\241 uma maneira r\303\241pida de resolver qualquer f\303\263rmula quadr\303\241tica. solve( a*x^2 + b*x + c = 0, x);

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Esta \303\251 a f\303\263rmula QuadForm := solve( a*x^2 + b*x + c = 0, x);

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Veja esse exemplo : QuadEq := x^2 + 9*x + 20 = 0;

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Agora, vamos substituir a, b e c - os coeficientes de x^2, x e do termo constante. Perceba que x^2 significa 1*x^2. subs( { a = 1, b = 9, c = 20}, {QuadForm });

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Podemos verificar nossa resposta usando o comando "solve" do Maple - e nesse casso \303\251 fator\303\241vel. solve( QuadEq, x); factor( lhs(QuadEq) );

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3*x^2 + 7*x + 4 ; simplify( subs( { a = coeff(%,x,2) , b = coeff(%,x,1), c = coeff(%,x,0) }, {QuadForm } ));

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Esse \303\251 um pequeno processo que identifica os coeficientes e encontra as solu\303\247\303\265es. Quadratic( 5*x^2 + 7*x + 4);

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Quadratic( 6*x^2 - 216 );

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Quadratic( x^2 + 7*x );

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Quadratic( x^2 + 12 );

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A express\303\243o D = 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 \303\251 chamada de discriminante de uma equa\303\247\303\243o quadr\303\241tica. Voc\303\252 pode reconhec\303\252-lo como a quantidade debaixo do radical na f\303\263rmula quadr\303\241tica. solve( a*x^2 + b*x + c = 0, x);

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Tamb\303\251m podemos expressar uma f\303\263rmula quadr\303\241tica assim: Discr := b^2 -4*a*c;

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x1 = (1/(2*a))*(-b + sqrt(Discr) ); x2 = (1/(2*a))*(-b - sqrt(Discr) );

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O discriminante nos diz o n\303\272mero e a qualidade das ra\303\255zes da equa\303\247\303\243o. Temos tres possibilidades: 1. D \303\251 positivo a equa\303\247\303\243o tem duas solu\303\247\303\265es distintas e reais 2. D \303\251 zero a equa\303\247\303\243o tem uma solu\303\247\303\243o real 3. D \303\251 negativo A equa\303\247\303\243o tem duas solu\303\247\303\265es complexas Agora vamos examinar exemplos de cada caso: Discriminant( x^2 + 3*x + 1);

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Discriminant( x^2 + 2*x + 1);

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Discriminant( x^2 + 1*x + 1);

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Discriminant( x^2 + 102*x + 1);

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a vantagem de usar o discriminante \303\251 que voc\303\252 pode saber se existem solu\303\247\303\265es reais ou n\303\243o, s\303\263 por olhar para o discriminante, em vez de ter todo o trabalho de resolver toda a f\303\263rmula quadr\303\241tica. O discriminante tamb\303\251m nos diz o n\303\272mero de interse\303\247\303\265es de x no gr\303\241fico. Temos tr\303\252s possibilidades: 1. D \303\251 positivo A equa\303\247\303\243o tem duas interse\303\247\303\265es de x 2. D \303\251 zero A equa\303\247\303\243o tem uma interse\303\247\303\243o de x 3. D \303\251 negativo A equa\303\247\303\243o nao tem interse\303\247\303\265es Vejamos o gr\303\241fico desses casos que acabamos de mencionar. plot( x^2 + x - 5 , x = -6..6);

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

plot( 8*x^2 + 40*x + 50 , x = -6..3, y = -5..100);

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Ig==

plot( x^2 + x + 2 , x = -4..3, y = 0..10);

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

O discriminante tamb\303\251m nos diz outra coisa. Uma express\303\243o quadr\303\241tica \303\251 fator\303\241vel se, somente se, o discriminante for um quadrado perfeito. Discriminant( (x+31)*(5*x-7) );

NiMtJS1EaXNjcmltaW5hbnRHNiMqJiwmJSJ4RyIiIiIjSiIiIiIiIiwmKiYiIiYiIiIlInhHIiIiIiIiIiIoISIiIiIi

sqrt( 26244);

NiMiJGki

Mas se somarmos um tornamos a express\303\243o infator\303\241vel. Discriminant( (x+31)*(5*x-7) + 1 );

NiMtJS1EaXNjcmltaW5hbnRHNiMsJiomLCYlInhHIiIiIiNKIiIiIiIiLCYqJiIiJiIiIiUieEciIiIiIiIiIighIiIiIiIiIiIiIiIiIiI=

sqrt( 26224);

NiMsJComIiIlIiIiKSIlUjsjIiIiIiIjIiIiIiIi

Tente voc\303\252 mesmo. Sempre que voc\303\252 come\303\247ar com uma express\303\243o quadr\303\241tica fator\303\241vel, o discriminante vai ser um quadrado perfeito. Discriminant( 8*x^2+62*x+117 );

NiMtJS1EaXNjcmltaW5hbnRHNiMsKComIiIpIiIiKSUieEciIiMiIiIiIiIqJiIjaSIiIiUieEciIiIiIiIiJDwiIiIi

sqrt( 100);

NiMiIzU=

Discriminant(102*x^2+145*x-408);

NiMtJS1EaXNjcmltaW5hbnRHNiMsKComIiQtIiIiIiklInhHIiIjIiIiIiIiKiYiJFgiIiIiJSJ4RyIiIiIiIiIkMyUhIiI=

sqrt(187489);

NiMiJEwl

Esses s\303\243o os passos que voc\303\252 pode usar para resolver a equa\303\247\303\243o \303\240 m\303\243o: 1. Leia a, b e c 2. Encontre o discriminante 3. Tire a raiz quadrada do discriminante 4. Forme duas fra\303\247\303\265es esse m\303\251todo \303\251 um pouco menor e menos sus\303\247et\303\255vel a erros do que simplesmente aplicar a f\303\263rmula. Exemplo 5.1: 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 . #1 Leia a, b, e c a:= 1; b:= 0; c:= -8; #2 Encontre o discriminate Discr := b^2 -4*a*c; #3 Tome a raiz quadrada do discriminante SqrtD := sqrt( Discr ); #4 forme as fra\303\247\303\265es x = (1/(2*a))*(-b + SqrtD ), (1/(2*a))*(-b - SqrtD );

NiM+JSJhRyIiIg==NiM+JSJiRyIiIQ==NiM+JSJjRyEiKQ==NiM+JSZEaXNjckciI0s=NiM+JSZTcXJ0REcsJComIiIlIiIiKSIiIyMiIiIiIiMiIiIiIiI=NiQvJSJ4RywkKiYiIiMiIiIpRicjRihGJ0YoRigsJEYmISIi

Exemplo 5.2: 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 . #1 Leia a, b, e c a:= 5; b:= 7; c:= -2; #2 Encontre o discriminate Discr := b^2 -4*a*c; #3 Tome a raiz quadrada do discriminante SqrtD := sqrt( Discr ); #4 Forme as fra\303\247\303\265es x = (1/(2*a))*(-b + SqrtD ), (1/(2*a))*(-b - SqrtD );

NiM+JSJhRyIiJg==NiM+JSJiRyIiKA==NiM+JSJjRyEiIw==NiM+JSZEaXNjckciIyopNiM+JSZTcXJ0REcqJCkiIyopIyIiIiIiIyIiIg==NiQvJSJ4RywmIyIiKCIjNSEiIiomRihGKSkiIyopIyIiIiIiI0YuRi4sJkYmRilGKkYp

Exemplo 5.3: 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 . #1 Leia a, b, and c a:= 12; b:= 24; c:= 6; #2 Encontre o discriminante Discr := b^2 -4*a*c; #3 Tome a raiz quadrada do discriminante SqrtD := sqrt( Discr ); #4 Forme as fra\303\247\303\265es x = (1/(2*a))*(-b + SqrtD ), (1/(2*a))*(-b - SqrtD );

NiM+JSJhRyIjNw==NiM+JSJiRyIjQw==NiM+JSJjRyIiJw==NiM+JSZEaXNjckciJClHNiM+JSZTcXJ0REcsJComIiM3IiIiKSIiIyMiIiIiIiMiIiIiIiI=NiQvJSJ4RywmIiIiISIiKiYiIiNGJylGKSNGJkYpRiZGJiwmRiZGJ0YoRic=