Question

a solução da equação x! sobre (x-2)! =30 (xEN e x maior ou igual 2)

Answer 1

$\dfrac{x!}{(x-2)!}=30\\\\\\\dfrac{x\cdot(x-1)\cdot(x-2)!}{(x-2)!}=30$

Cancelando (x - 2)!:

$x\cdot(x-1)=30\\x^{2}-x=30\\x^{2}-x-30=0\\\\\Delta=b^{2}-4c\\\Delta=(-1)^{2}-4\cdot1\cdot(-30)\\\Delta=1+120\\\Delta=121\\\\x=\dfrac{-b\pm\sqrt{\Delta}}{2a}=\dfrac{-(-1)\pm\sqrt{121}}{2\cdot1}=\dfrac{1\pm11}{2}$

Achando as raízes:

$x'=\dfrac{1+11}{2}=6\\\\\\x''=\dfrac{1-11}{2}=-5$

x = -5 não serve, logo:

$\boxed{\boxed{x=6}}$

Answer 2

$\dfrac{x!}{(x-2)!} =30$      $(x\in\text{N e }x \geq 2)$

$\dfrac{x.(x-1).(x-2)!}{(x-2)!} =30$

$x.(x-1)=30$

$x^{2} -x=30$

$x^{2} -x-30=0$

$(x+5).(x-6)=0$

$x+5=0$  ⇒  $x=-5$  →  não serve, pois $x \geq 2$
ou
$x-6=0$  ⇒  $x=6$

$S=\{6\}$