Question

ajutor va rog mult!!!​

Answer 1

a)

$(x+2)(x+5) = x^{2} +5x+2x+10=x^{2} +7x+10$

b)

$(x+3)(x+4) = x^{2}+4x+3x+12 = x^{2}+7x+12$

notam:

$t = x^{2}+7x+10$

$x^{2}+7x+12 = x^{2}+7x+10+2 = \red{ \bf t+2}$

$E(x) = (x+2)(x+3)(x+4)(x+5)+1 = \Big[(x+2)(x+5)\Big] \cdot \Big[(x+3)(x+4)\Big] + 1 = (x^{2}+7x+10) \cdot (x^{2}+7x+12) + 1$putem scrie:

$E(t) = t(t+2)+1 = t^{2}+2t+1 = (t+1)^{2}$

revenim la notatie:

$E(x) = (x^{2}+7x+10)^{2}$

atunci:

$E(n) = (n^{2}+7n+10)^{2}$

$n \in \mathbb{N} \iff n \geq 0 \implies \Big|n^{2}+7n+10\Big| = n^{2}+7n+10$

$\sqrt{E(n)} = \sqrt{(n^{2}+7n+10)^{2}} = n^{2}+7n+10$

$\implies \sqrt{E(n)} \in \mathbb{N}$

q.e.d.