Question

se considera expresia E(x)=(2x-1)^2-3(x-1)^2 unde x este nr real.demonstratrati caE(-a)=E(a-2)oricare ar fi nr real a.​

Answer 1

Explicație pas cu pas:

$E(x) = (2x-1)^{2}-3(x-1)^{2} = \\ = 4 {x}^{2} - 4x + 1 - 3 {x}^{2} + 6x - 3$

$= {x}^{2} + 2x - 2$

$E(- a) = {(- a)}^{2} + 2(- a) - 2 =$

$= {a}^{2} - 2a - 2$

$E(a - 2) = {(a - 2)}^{2} + 2(a - 2) - 2 = \\ = {a}^{2} - 4a + 4 + 2a - 4 - 2$

$= {a}^{2} - 2a - 2$

$\implies E(- a) = E(a - 2)$

$q.e.d.$

Answer 2

$E(x) = (2x - 1) {}^{2} - 3(x - 1) {}^{2} \\ E(x) = 4x {}^{2} - 4x + 1 - 3(x {}^{2} - 2x + 1) \\ E(x) = 4x {}^{2} - 4x + 1 - 3x {}^{2} + 6x - 3 \\ E(x) = x {}^{2} + 2x - 2 \\ \\ \boxed{ \text{demonstratie : }}\\ E( - a) = a {}^{2} - 2a - 2 \\ E(a - 2) = a {}^{2} - 4a + 4 + 2a - 4 - 2 \\ E(a - 2) = a {}^{2} - 2a - 2 \\ \implies a {}^{2} - 2a - 2 = a {}^{2} - 2a - 2 \\ \implies \: a {}^{2} - 2a = a {}^{2} - 2a \\ \implies \: a {}^{2} = a {}^{2} \\ \implies \: 0 = 0 \\ \iff \: a\in\mathbb{R}$