Question

Exercițiul 40 vă rog rapid dau multe puncte

Answer 1

a)

$x^2+2x-3=x^2-x+3x-3=x(x-1)+3(x-1)=(x-1)(x+3),\forall x\in\mathbb{R}$

b)

$x^2+4x+3=x^2+x+3x+3=x(x+1)+3(x+1)=(x+1)(x+3)\\\\\displaystyle E(x)=\left(\frac{x^2+4x+3}{x^2+2x-3}-\frac2{x+1}+\frac7{1-x^2}\right)\cdot(x^2-1)=\left(\frac{(x+1)(x+3)}{(x+3)(x-1)}-\frac2{x+1}-\frac7{x^2-1}\right)\cdot(x^2-1)\\\text{Facem simplific\u ari \c si apoi aducem la acela\c si numitor }(x+1)(x-1)=x^2-1\\\\\displaystyle E(x)=\frac{(x+1)^2-2(x-1)-7}{x^2-1}\cdor(x^2-1)=x^2+2x+1-2x+2-7=x^2-4=(x+2)(x-2),\forall x\in\mathbb{R}$

c)

$E(a)=a-4\\\text{Folosim punctul b)}\\(a+2)(a-2)=a-4\Rightarrow a^2-4=a-4\Rightarrow a^2=a\Rightarrow a^2-a=0\Rightarrow a(a-1)=0\Rightarrow a=0\text{ sau } a-1=0\Rightarrow a=0\text{ sau }a=1\Rightarrow a\in\{0,1\}$