Question

Consideram expresia E(x)=x la patrat +4x+5, x€R. a)Aratati ca E(x)>sau egal 1,pentru orice x€R. b)Determinati numerele reale a,b,c, stiind ca E(a)+E(b)+E(c)=3​

Answer 1

Răspuns:

Explicație pas cu pas:

E(x) = x²+4x+5

(x+2)² = x²+4x+4  =>

E(x) = x²+4x+4+1 = (x+2)²+1

(x+2)² ≥ 0 ; (∀) x ∈ R  =>

E(x) ≥ 1  ; (∀) x ∈ R

E(a) + E(b) + E(c) = 3 <=>

a²+4a+5+b²+4b+5+c²+4c+5 = 3  <=>

(a+2)²+1 + (b+2)²+1 + (c+2)² + 1 = 3  <=>

(a+2)²+(b+2)²+(c+2)² = 0  =>

(a+2)² = 0  ; (b+2)² = 0 ; (c+2)² = 0 =>

a = b = c = -2