Question

Se considera expresia E(x)=(x²-x+3)²-(x²-x)²-5x²,unde x este nr. real . Demonstrati ca E(n) este patrat perfect , oricare ar fi nr. natural ,,n"

Answer 1

[tex]E(n)=(n^2-n+3)^2-(n^2-n)^2-5n^2\\ E(n)=(n^2-n+3-n^2+n)(n^2-n+3+n^2-n)-5n^2\\ E(n)=3(2n^2-2n+3)-5n^2\\ E(n)=6n^2-6n+9-5n^2\\ E(n)=n^2-6n+9\\ E(n)=(n-3)^2\ \forall n\in N[/tex]