ࡱ> kmj7 A6bjbjUU 7|7|l8.l,Y"!@@@WWWWWWW$Y [|X@<@@@XUM! XUMUMUM@0!WUM@WUMUMN:/S,zS! `OVE[S S<X0YeSRb\Jb\SUMECUAbIA DE GRADUL AL II-LEA Fie problema: O cas are baza n form de dreptunghi, cu lungimea de 13m _i lcimea de 7,5m. Proprietarul dore_te s-_i construiasc o bordur de ciment, de aceea_i lcime pe toate laturile casei (vezi desenul). Fondurile pe care le are l oblig la o suprafac construibil de 33m2.  x 13m n condiciile date, care este lcimea maxim pe care o poate avea bordura casei?  CASA 7,5m Pentru rezolvarea acestei probleme notm cu x, n metri, lcimea bordurii _i putem scrie urmtoarea ecuacie: 4x2 + 41x =33 ( 4x2 + 41x  33 = 0 Se observ c aceast ecuacie este diferit de tipul de ecuacii nvcate anterior. Deoarece necunoscuta x apare _i la puterea a doua, aceast ecuacie spunem c se nume_te de gradul al II-lea. Forma general a unei ecuacii de gradul al II-lea este: ax2 + bx + c = 0 (1) unde a,b,c sunt numere reale, cu a ( 0. Aceast ecuacie se nume_te de gradul al II-lea cu coeficienci reali. Rezolvarea ecuaciei (1) presupune determinarea tuturor soluciilor (rdcinilor) sale. Existenca rdcinilor reale precum _i numrul lor depind de expresia b2  4ac (2) care se nume_te discriminantul ecuaciei de gr. al II-lea _i se noteaz cu (. Dac discriminantul este pozitiv, atunci ecuacia are dou rdcini reale, diferite ntre ele: (3) n cazul n care ( = 0, atunci ecuacia are dou solucii reale, egale: Putem avea _i dou cazuri particulare de rezolvare a ecuaciei (1) _i anume: a)Dac coeficientul b al lui x este nul atunci ecuacia devine: ax2 + c = 0 n aceast situacie ecuacia are dou solucii reale, egale numai dac c ( 0 _i ele sunt: b)Dac termenul liber c este egal cu zero. atunci forma ecuaciei este:  ax2 + bx = 0 Rezolvarea este: Ecuacia de gradul al doilea, care are discriminantul ( ( 0, admite _i dou forme particulare importante, _i anume:  1. Dac n ecuacia (1) coeficientul b al lui x este de forma: b = 2b1 atunci obcinem: ax2 + 2b1x + c = 0, pentru care discriminantul devine  iar rdcinile vor fi de forma . 2. Forma redus a ecuaciei de gradul al doilea. O ecuacie de gradul al doilea se nume_te redus dac coeficientul lui x2 = 1. Forma general a ecuaciei reduse este: x2 + px + q = 0, unde p, q sunt numere reale.  Dac n relaciile (1), (2), (3) nlocuim a, b, c respectiv cu 1, p, q vom obcine formula pentru rdcinile ecuaciei de gradul al doilea sub form redus:  ntre coeficiencii _i rdcinile unei ecuacii de gr. al II-lea (1) se poate stabili un set de relacii cu aplicacie practic:   (4) Relaciile (4) poart denumirea de Relaciile lui Vite. Cu aceste relacii se poate deci calcula suma _i produsul rdcinilor reale ale ecuaciei (1) fr a le afla efectiv. s = x1 + x2 , p = x1 ( x2 (5) Aceste relacii ne permit s formm o ecuacie de gr. al II-lea atunci cnd cunoa_tem rdcinile, astfel: x2  sx + p = 0 De utilitate practic mai este _i studiul semnelor rdcinilor unei ecuacii de gr al II-lea, mai ales cnd aceasta este cu parametru. Acest lucru se poate face studiind semnul discriminantului, sumei _i produsului rdcinilor din relacia (2), respectiv din relaciile lui Vite (4). Se poate construi urmtorul tabel: (<0Ecuacia (1) nu are rdcini reale.((0p>0s>0 x1>0, x2>0s<0 x1<0, x2<0 p<0s>0 x1>0, x2<0, (x1(>(x2(s<0 x1>0, x2<0, (x1(<(x2( Observacii: 1. Fie s = 0 . Ecuacia are rdcini reale numai dac p ( 0. n acest caz avem x1 +x2 = 0 adic x1 = -x2 . 2. Fie p = 0 . Atunci x1 = 0 _i x2 = s. APLICAbII S rezolvm ecuacia problemei din introducerea lucrrii: 4x2 + 41x  33 = 0 ( = 412  4( 4 ( ( - 33) = 1681 + 528 = 2209  aceast solucie nu este acceptabil din punctul de vedere al problemei pentru c este negativ. Deci bordura casei va avea lcimea maxim de 0,75m. 2. S se studieze natura rdcinilor ecuaciei mx2 +(m  1)x  (m  2) = 0 n funccie de parametrul real m. Vom calcula _i vom studia, mai nti, semnul pentru (, s, _i p. (= (m  1)2 + 4m(m  2)= m2  2m +1 +4m2  8m = 5m2  10m +1 ( va fi negativ ntre valorile m1 _i m2 _i pozitiv n rest.  -( 0 1 +(1  m+ + + + + + + + + + + + + 0 - - - - - - - -  m- - - - - - - 0 + + + + + + + + + + + + + s- - - - - - - / + + + + + + 0 - - - - - - - - -   - ( 0 2 +(2 m+ + + + + + + + + + + + + 0 - - - - - - - - - m- - - - - - - 0 + + + + + + + + + + + + +  p- - - - - - - - / + + + + + + 0 - - - - - - - - -  m(spnatura rdcinilor( -( ; 0 )+--( x1(x2(R, x1(0, x2(0, (x1(((x2(0+//Ec de gr I , x  2 = 0, x = 2+++( x1(x2(R, x1(0, x2(00++( x1=x2(R+-++Ecuacia dat nu are solucii reale.1-0+Ecuacia dat nu are solucii reale.--+Ecuacia dat nu are solucii reale.0-+( x1=x2(R-- +-+( x1(x2(R--2+-0( x1(x2(R, x1=0, x2(0( 2; +( )+--( x1(x2(R, x1(0, x2(0, (x1(((x2( PAGE  PAGE 5  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  :tv|~28<@|~hln~  *  ؽرؽؽؽاررر}n^n56CJH*OJQJ\]^J56CJOJQJ\]^J>*CJOJQJ^J$ j6CJOJQJ]^JmH sH 6CJH*OJQJ]^JCJOJQJ^Jy(6CJOJQJ]^J jOJQJU^JmHnHuCJH*OJQJ^JCJOJQJ^J5>*CJ OJQJ\^J!5>*CJ OJQJ\^JmH sH "8:<Z|$2:$ Ddh$&#$/Ifa$ $dh`a$ x$dha$$a$ 83j3@6:<BDF{{{$ Ddh$&#$/Ifa$h$$Ifl,hD 606 4 laFHJLprrXWWWW$ Ddh$&#$/Ifa$$$IflFh ~  606    4 lartvxzpUUU$ Ddh$&#$/Ifa$$$IflFh ~  606    4 laz|h. d  pd\\\\\\\\$dha$ $ dha$$$IflFh ~  606    4 la  &  (*.2DFVXLN׼ם׌׌׀tttgttXtt׌t j6CJOJQJ]^J6CJH*OJQJ]^J6CJOJQJ]^J jDCJOJQJ^J jOJQJU^JmHnHu jD5CJOJQJ\^J56CJH*OJQJ\]^J56CJOJQJ\]^J jCJOJQJ^JCJOJQJ^JCJOJQJ^JmH sH $56CJOJQJ\]^JmH sH " .|~L $ hdha$ $dh^a$ $dh`a$$a$$dha$\^`b&(8:Zfh,.0Jz~@T^`hlnݴݴݴݜݴݴݴݴjUmHnHu6CJH*OJQJ]^J jOJQJU^JmHnHu jCJOJQJ^J jDCJOJQJ^JCJOJQJ^J6CJOJQJ]^J6CJH*OJQJ]^J8T,. 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