Question

Aratati ca expresia (x² +2x + 2 )( x² +2x +4)+1 este pătrat perfect , oricare ar fi numarul real x .

Answer 1

$E=(x^{2}+2x+2)(x^{2}+2x+4)+1=(x^{2}+2x+1+1)(<span>x^{2}+2x+1+3</span>)+1=((x+1)^{2}+1)((x+1)^{2}+3)+1$
Notam
$(x+1)^{2}=a$
$E=(a+1)(a+3)+1=a^{2}+3a+a+3+1=a^{2}+4a+4=(a+2)^{2}$
Deci E este patrat perfect oricare ar fi x

Answer 2

[tex]E=(x^2+2x+2)(x^2+2x+4)+1 (patrat\ perfect)\\ Notam: x^2+2x=a\\ E=(a+2)(a+4)+1\\ E=a^2+4a+2a+8+1\\ E=a^2+6a+9\\ E=(a+3)^2\\ E=(x^2+2x+3)^2\\ Asadar\ e\ patrat\ perfect\ pentru\ orice\ x\in R.[/tex]