Question

Rezolvati in multimea numerelor reale ecuatia $\sqrt{x^2+x+2}$ = x^2+x.

Answer 1

Explicație pas cu pas:

V : radical

V(x²+x+2)=x²+x

x²+x+2=x⁴+2x³+x²

x+2=x⁴+2x³

x+2-x⁴-2x³=0

x+2-x³(x+2)=0

(x+2)(-x³+1)=0

______

x+2=0 | x=-2

-x³+1=0 | x=1

___

S={-2,1}

Answer 2

Răspuns:

S={-2;1}

Explicație pas cu pas:

C.E

x²+x+2≥0

adevarat ∀x∈R, pt ca Δ<0

x²+x≥0...x∈(-∞;-1]∪[0;∞)

rezolvare

ridicam la patrat

x²+x+2=x^4+2x³+x²

x^4+2x³-(x+2)=0

x³(x+2)-(x-2)=0

(x-2) (x³-1)=0

(x+2)(x-1) (x²+x+1)=0

(x²+x+1)=0 nu are radacini reale pt ca Δ<0

raman

x+2=0⇒x1=-2 ∈Domeniului de existenta

x-1=0⇒x2=1 ∈Domeniului de existenta

Verificare

pt -2, avem √4=4-2

si respectiv, pt x=1

√4=1+1

adevarat, bine rezolvat