Question

Fie expresia E(a)=(2a/a+1 - 2/a^2-1 + a/a-1) ÷ 9a^2-4/9a^2-6a
1) aflat a pt.care E(a)are sens
2) arătat ca E(a)= 3a/a+1

Answer 1

expresia are sens daca numitorii nu se anuleaza

a+1≠= a≠ -1

a-1≠0 a≠1

9a²-6a≠0 a(3a-2)≠0 a≠0 si a≠2/3

aceste conditii dau si a²-1≠0

E are sens pentru a∈R-{-1, 0, 2/3, 1}

E(a)=[2a(a-1)-2+a(a+1)]/(a²-1):(3a-2)(3a+2)/3a(3a-2)=(2a²-2a-2+a²+a)/(a-1)(a+1):(3a+2)/3a=(3a²-a-2)/(a-1)(a+1):(3a+2)/3a=(3a+2)(a-1)/(a-1)(a+1) * 3a/(3a+2)=3a/(a+1)

am descompus pe f(x)=3a²-a-2 afland mai intai radacinile x1= -1 si x2= -2/3 si folosind regula f(x)=3(x-x1)(x-x2)

am mai folosit diferenta de patrate

a²-c²=(a-c)(a+c)

si regula de impartire a doua fractii (a/b)/(c/d)=a/b * d/c (deci se inmulteste prima cu a doua inversata