Question

Determine o valor de n, sabendo que (n!)2 - 126n! + 720 = 0

Answer 1

Resposta:

n=3 ou n=5

Explicação passo-a-passo:

(n!)²-126(n!)+720=0

Chamando x=(n!)

x²-126x+720=0

$\displaystyle Aplicando~a~f\'{o}rmula~de~Bhaskara~para~x^{2}-126x+720=0~~e~comparando~com~(a)x^{2}+(b)x+(c)=0,~determinamos~os~coeficientes:~\\a=1{;}~b=-126~e~c=720\\C\'alculo~do~discriminante~(\Delta):&\\&~\Delta=(b)^{2}-4(a)(c)=(-126)^{2}-4(1)(720)=15876-(2880)=12996\\C\'alculo~das~raizes:&\\x^{'}=\frac{-(b)-\sqrt{\Delta}}{2(a)}=\frac{-(-126)-\sqrt{12996}}{2(1)}=\frac{126-114}{2}=\frac{12}{2}=6\\\\x^{''}=\frac{-(b)+\sqrt{\Delta}}{2(a)}=\frac{-(-126)+\sqrt{12996}}{2(1)}=\frac{126+114}{2}=\frac{240}{2}=120$Para (n!)=x'=6

n!=6

3.2.1=6

3!=6

n=3

Para (n!)=x''=120

n!=120

5.4.3.2.1=120

5!=120

n=5