Răspuns:
a) (-∞;0)∪(1;+∞)
d) 194
Explicație pas cu pas:
a) f∩y=(0;-1) => f=-1 => x²+2xm+m-1=-1 /+1
x²+x*2m+m-1+1=0
x²+x*2m+m=0
Intersecteaza in doua puncte distincte => ecuatia are doua radacini distincte. Ecuatia are 2 radacini distincte atunci cand Δ>0.
Δ=b²-4ac=(2m)²-4*1*m=4m²-4m=4(m²-m)
Δ>0 => 4(m²-m)>0 => m²-m>0
Acum trebuie sa facem tabelul de semn al ecuatiei de gradul 2: m²-m=0
m(m-1)=0 => Radacinile sunt: m=1 si m=0
m |-∞...........0............1.........+∞
m²-m |+++++++0---------0++++++
=> m∈(-∞;0)∪(1;+∞)
=> Graficul functiei intersecteaza dreapta y=-1 in doua puncte distincte
∀m∈(-∞;0)∪(1:+∞)
d) x²+4x+1=0
Δ=b²-4ac=4²-4*1*1=16-4=12
x=-b±√Δ/2a=-4±√12/2=4±2√3/2=-2±√3
=> x₁=-2+√3 =>
[tex]x_{1}^{2}=(2+\sqrt{3})^{4}=(2+\sqrt{3})^{2} (2+\sqrt{3})^{2}[/tex]
[tex](4+2\sqrt{3}+3)^{2}=(7+4\sqrt{3})^{2}=(49+4*7*2\sqrt{3}+16*3)=(49+28*2\sqrt{3}+48)[/tex]
[tex]=97+56\sqrt{3}[/tex]
Aidoma lui x₂⁴=-2-√3, o sa vina [tex]97-56\sqrt{3}[/tex]
=> x₁⁴+x₂⁴=[tex]97-56\sqrt{3}+97+56\sqrt{3}=97*2=194[/tex]
