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Question Number 84515 by M±th+et£s last updated on 13/Mar/20

$$Q.solve$$$$x^3-x=x!$$

Answered by mr W last updated on 13/Mar/20

$$(x+1)x(x-1)=x(x-1)(x-2)!$$$$x\ne 0,x\ne 1$$$$\Rightarrow x+1=(x-2)!$$$$lett=x-2$$$$\Rightarrow t+3=t!$$$$\Rightarrow t=3$$$$\Rightarrow x=3+2=5$$

Commented by mr W last updated on 14/Mar/20

$$att=2:$$$$t+3=5,t!=2\Rightarrow t+3>t!$$$$att=4:$$$$t+3=7,t!=24\Rightarrow t+3<t!$$$$thatmeansbetweent=2andt=4there$$$$isapointfort+3=t!.wecancheck$$$$witht=3:t+3=6=t!=6.i.e.t=3is$$$$asolutionandtheonlysolution,$$$$becauseupont=4,t!\gg t+3.$$

Commented by M±th+et£s last updated on 14/Mar/20

$$thankyousir$$$$buthowdidyougetthat$$$$t+3=t!$$$$t=3??$$

Commented by M±th+et£s last updated on 14/Mar/20

$$godblessyousir$$

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