Question

sa se determine m apartine lui R pentru care ecuatia x^2 - mx+2=0 are doua radacini reale si distincte

Answer 1

Daca ecuatia are doua radacini reale distincte, atunci conditia este:

Δ>0

x²-mx+2=0

Δ=(-m)²-4*1*2

Δ=m²-8

m²-8>0

Atasam ecuatia:

m²-8=0

m²=8

m1=-2√2

m2=2√2

Si facem tabel de semn:

_m__ |-inf________-2√2________2√2_________inf

m²-8_|++++++++++++++0-------------------0++++++++++++++

m²-8>0 => m∈(-inf;-2√2)∪(2√2;inf).

Answer 2

O ecuație de gradul al 2-lea are rădăcini reale distincte dacă Δ >0.

$\it \Delta = m^2 - 8 >0 \Rightarrow m^2 > 8 \Rightarrow \sqrt{m^2} >\sqrt8 \Rightarrow \sqrt{m^2} >\sqrt{4\cdot2} \Rightarrow \\ \\ \\ \Rightarrow |m| >2\sqrt2 \Rightarrow m\in(-\infty,\ -2\sqrt2) \cup(2\sqrt2,\ \infty).$