Question

ESTE URGENT
problema 33 de la pagina 89 din culegerea Mate2000 ​

Answer 1

Explicație pas cu pas:

a)

$E(x)=(2x+3)^2-(x-3)(2+x)-(x+2)^2-x(x+7)-10, x\in \mathbb{R}\\E(x)=4x^2+12x+9-(2x+x^2-6-3x)-(x^2+4x+4)-x^2-7x-10=\\4x^2+12x+9+x-x^2+6-x^2-4x-4-x^2-7x-10=\\x^2+2x+1=(x+1)^2 \\E(x)=(x+1)^2 \ \forall x\in\mathbb{R}$

b)

$E(a^2)-E(-a^2)=16$

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

$(a^2+1)^2-(-a^2+1)^2=16\\\boxed{a^2-b^2=(a+b)(a-b)}\\(a^2+1-a^2+1)(a^2+1+a^2-1)=16\\2 \cdot 2a^2=16\\a^2=4 \implies a=\pm2 \in \mathbb{R}$