Question

se considera expresia a(x)= (2x+1)^2-(2x+2)^2-(2x+3)^2+(2x+4)^2 x eR. Aratati ca A(n) este patrat perfect

Answer 1

$a(x)= \\ \\ =(2x+1)^2-(2x+2)^2-(2x+3)^2+(2x+4)^2\\ \\ = (2x+1)^2-(2x+3)^2+(2x+4)^2 - (2x+2)^2 \\ \\ = \Big((2x+1)-(2x+3)\Big)\Big((2x+1)+(2x+3)\Big) + \\ +\Big((2x+4)-(2x+2)\Big)\Big((2x+4)+(2x+2)\Big) \\ \\ = (-2)(4x+4)+2(4x+6) \\ \\ = 2(-4x-4+4x+6) \\ \\ = 2\cdot 2 \\ \\ = 4 \rightarrow patrat~perfect$