Question

Sa se rezolve problema urmatoare.

Answer 1

a.  Fractia F=x² +2mx +3m-2 /x² +8x+m are sens daca si numai daca x² +8x+m≠0 .

Pentru x² +8x+m=0 => Δ=b² -4ac=8² -4·1·m=64-4m dar Δ>=0 <=> 64-4m >= 0 <=> 16-m >= 0 <=> m <= 16 dar deoarece m∈R => m∈(-∞,16] .

Asadar fractia de mai sus are sens pentru m∈(16,∞) .

b.  x² +8x+m > 0 => a=1 > 0 si Δ < 0 <=> 64-4m < 0 <=> 16-m < 0 <=> m > 16 <=> m∈(16,∞) .

c.  x² +2mx+3m -2 > 0 <=> a=1 > 0 si Δ < 0 <=> b² -4ac=4m² -4·1·(3m-2)=4m² -12m +8=4m² -12m +8 < 0 <=> m² -3m +2 < 0 <=> m² -m-2m+2 < 0 <=>

m·(m-1)-2·(m-1) < 0 <=> (m-1)·(m-2) < 0 => m-1 > 0 => m > 1 si m-2 < 0 => m < 2 =>m∈(1,2) dar m∈(16,∞) => nu convine sau m-1 < 0 => m < 1 si m-2 > 0 => m > 2 care este imposibil ,deci nu avem solutii .