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Ecua?ia de gradul I
ax+b=0
cazI a?0 ? x=-b/a - ecua?ie compatibila determinata
cazII a=0 ? ? b=0 ? x ?R - ecua?ie compatibila nedeterminat?
? b?0 ? x ? R - ecua?ie incompatibila
Ecua?ii - compatibile = au cel pu?in o r?d?cina
- incompatibile = nu au r?d?cina in domeniu
Ec. compatibile - determinate = au o r?d?cina unica
- nedeterminate = au cel pu?in doua r?d?cini
Ecua?ia de gradul al doilea
ax2+bx+c=0 a, b, c ? R a?0
?=b2-4ac
1) ?>0 ? x1, x2 ? R, x1?x2 x1,2=(-b???)/2a
2) ?=0 ? x1, x2 ? R, x1=x2=-b/2a
3) ?<0 ? ecua?ia nu are r?d?cini reale
Daca b este par ?b=2b1 atunci ?= b2-4ac=4(b12-ac)=4 ?1 ?1= b12-ac
1) ?1>0 ?x1,2=(-b1???1)/a
2) ?1=0 ?x1=x2=-b1/a
3) ec. nu are r?d?cini reale
In general
S=x1+x2=(-b+??-b-??)/(2a)=-b/a
P=x1x2=[(-b+??)(-b-??)]/(4a2)=c/a
Concluzie
-Rela?iile lui Viéte
S= x1+x2=-b/a
P= x1x2=c/a
Reciproc: scrierea ecua?iei de gradul al doilea când se cunosc r?d?cinile
ax2+bx+c=o ? :a
x2-(-b/a)x+c/a=0
x2-Sx+P=0
De ex:
X1=1 ? S=1-2=-1
X2=-2 P=1(-2)=-2
x2-(-1)x+(-2)=0 x2+x-2=0
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