Question

resolva as equações a seguir n! (n-2)!=42

Answer 1

n.(n-1).(n-2)! / (n-2)! = 42

n²-n = 42

n²-n-42 = 0

delta = (-1)² - 4.1.(-42)

delta = 1 + 168

delta = 169

 

n = [-(-1)+/-\/169] / 2

n = (1+/-13)/2

n' = 14/2 = 7

n" = -12/2 = -6 (não convém)

Answer 2

Explicação passo-a-passo:

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$\sf{ \frac{n \times (n - 1) \times (n - 2)!}{(n - 2)!} = 42 } \\ \\ \sf{ \frac{(n - 2)! \times (n - 1)n}{(n - 2)!} = 42 } \\ \\ \sf{ \frac{\cancel{\blue {(n - 2)! \times (n - 1)n}}}{(n - 2)!} = 42} \\ \\ \sf{ \frac{n!}{(n - 2)!} = 42} \\ \\ \sf{ \frac{n \times (n - 1) \times (n - 2)!}{(n - 2)!} = 42 } \\ \\ \sf{ \frac{n \times (n - 1) \times \cancel{\blue {(n - 2)!}}}{\cancel{\blue {(n - 2)!}}} = 42} \\ \\ \sf{n \times (n - 1) = 42} \\ \\ \sf{n \times n - 1 = 42} \\ \\ \sf{ {n}^{2} - n = 42} \\ \\ \sf{ {n}^{2} - n - 42 = 0} \\ \\ \sf{ {n}^{2} + 6n - 7n - 42 = 0 } \\ \\ \sf{n(n + 6) - 7(n + 6) = 0} \\ \\ \sf{(n + 6) \times (n - 7) = 0} \\ \\ \sf{n + 6 = 0} \\ \sf{n - 7 = 0} \\ \\ \sf{n = - 6} \\ \sf{n = 7} \\ \\ \red{ \boxed{ \green{ \boxed{ \pink{ \boxed{ \sf{n = 7}} }} } } }$