Question

calcula equacao binominais 

A)   x        =55x     
3!(x-3)!          3

b) n!    = 1
(n+2)!    30

Answer 1

$\frac{x!}{3!(x-3)!}=\frac{55x}{3}\\\\\frac{x(x-1)(x-2)(x-3)!}{3.2.1.(x-3)!}=\frac{55x}{3}\\\\\frac{x(x-1)(x-2)}{6}=\frac{55x}{3}\\\\\frac{x(x-1)(x-2)}{2}=\frac{55x}{1}\\\\\frac{(x-1)(x-2)}{2}=\frac{55}{1}\\\\(x-1)(x-2)=2.55\\x^{2}-2x-x+2=110\\x^{2}-3x+2-110=0\\x^{2}-3x-108=0$

$\Delta=b^{2}-4ac\\\Delta=(-3)^{2}-4.1.(-108)\\\Delta=9+432\\\Delta=441$

$x=(-b\pm\sqrt{\Delta})/2a\\x=(-[-3]\pm\sqrt{441})/(2.1)\\x=(3\pm21)/2\\\\x'=(3+21)/2\\x'=24/2\\x'=12\\\\x''=(3-21)/2\\x''=-18/2\\x''=-9$

Como só calculamos fatorial de números naturais, descartamos x = - 9

$S=\{12\}$

b)

$\frac{n!}{(n+2)!}=\frac{1}{30}\\\\\frac{n!}{(n+2)(n+1)n!}=\frac{1}{30}\\\\\frac{1}{(n+2)(n+1)}=\frac{1}{30}\\\\(n+2)(n+1)=30\\n^{2}+n+2n+2=30\\n^{2}+3n+2=30\\n^{2}+3n+2-30=0\\n^{2}+3n-28=0$

$S=-b/a=-3/1=-3\\P=c/a=-28/1=-28$

Raízes: 2 números que quando somados dão - 3 e quando multiplicados dão - 28

$n'=4\\n''=-7$

Descartamos n = - 7

$S=\{4\}$