Question

Fie expresia:E(x)=(2x+1)la puterea a 2a -(x-1)la puterea a 2a +(x-2)(x+2)-3x la puterea a 2a +14,unde x apartine lui R.
a)Aratati ca E(x)=x la a 2a +6x+10,pentru orice x apartine lui R.
b)Aratati ca E(x)-2=(x+2)(x+4.
c)Aratati ca E(x)mai mare decat 0,pentru orice x apartine lui R.

Answer 1

$\displaystyle E(x)=(2x+1)^2-(x-1)^2+(x-2)(x+2)-3x^2+14 \\ a).E(x)=(2x)^2+2 \cdot 2x \cdot 1+1^2-(x-1)^2+x^2-4-3x^2+14 \\ E(x)=4x^2+4x+1-(x^2-2 \cdot x \cdot 1+1^2)+x^2-4-3x^2+14 \\ E(x)=4x^2+4x+1-(x^2-2x+1)+x^2-4-3x^2+14 \\ E(x)=4x^2+4x+1-x^2+2x-1+x^2-4-3x^2+14 \\ E(x)=x^2+6x+10 \\ b).E(x)-2=x^2+6x+10-2 \\ E(x)-2=x^2+6x+8 \\ E(x)-2=x^2+2x+4x+8 \\ E(x)-2=x(x+2)+4(x+2) \\ E(x)-2=(x+2)(x+4) \\ c).E(x)\ \textgreater \ 0 \\ x^2+6x+10\ \textgreater \ 0 \\ (x+3)^2+1\ \textgreater \ 0 \\ (x+3)^2\ \textgreater \ -1 \Rightarrow Adevarat~\forall x \in R$