Question

Решить комбинаторное уравнение

Answer 1

$3C_{x+1}^{x-1}+4C_{x}^{x-2} =418\\\\3*\frac{(x+1)!}{(x-1)!*(x+1-x+1)!}+4*\frac{x!}{(x-2)!*(x-x+2)!}=418\\\\3*\frac{(x+1)!}{(x-1)!*2!}+4*\frac{x!}{(x-2)!*2!} =418\\\\3*\frac{(x-1)!*x*(x+1)}{(x-1)!*1*2}+4*\frac{(x-2)!*(x-1)*x}{(x-2)!*1*2}=418\\\\3x(x+1)+4x(x-1)=836\\\\3x^{2}+3x+4x^{2}-4x-836=0\\\\7x^{2} -x-836=0\\\\D=(-1)^{2}-4*7*(-836)=1+23408=23409=153^{2} \\\\x_{1}=\frac{1+153}{14} =\frac{154}{14}=11\\\\x_{2}=\frac{1-153}{14} =-\frac{152}{14}=-10\frac{6}{7} -<0 -neyd$

$Otvet:\boxed{x=11}$