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Question Number 171070 by MathsFan last updated on 07/Jun/22

x^2 −1=2^x find x

$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{1}=\mathrm{2}^{\boldsymbol{\mathrm{x}}} \\ $$$$\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{x}} \\ $$

Answered by Rasheed.Sindhi last updated on 07/Jun/22

x^2 −1=2^x ;find x Assuming x a whole number: (x−1)(x+1)=2^x Both x−1 & x+1 are powers of 2 and have difference 2 between them. Only 2 & 4 are such numbers x−1=2 ∧ x+1=4⇒x=3 3^2 −1=2^3 ✓ x=3

$$\mathrm{x}^{\mathrm{2}} −\mathrm{1}=\mathrm{2}^{\mathrm{x}} \:;\mathrm{find}\:\mathrm{x} \\ $$$${Assuming}\:\mathrm{x}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{number}: \\ $$$$\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)=\mathrm{2}^{\mathrm{x}} \\ $$$$\mathrm{Both}\:\mathrm{x}−\mathrm{1}\:\&\:\mathrm{x}+\mathrm{1}\:\mathrm{are}\:\mathrm{powers}\:\mathrm{of}\:\mathrm{2} \\ $$$$\mathrm{and}\:\mathrm{have}\:\mathrm{difference}\:\mathrm{2}\:\mathrm{between}\:\mathrm{them}. \\ $$$$\mathrm{Only}\:\mathrm{2}\:\&\:\mathrm{4}\:\mathrm{are}\:\mathrm{such}\:\mathrm{numbers} \\ $$$$\mathrm{x}−\mathrm{1}=\mathrm{2}\:\wedge\:\mathrm{x}+\mathrm{1}=\mathrm{4}\Rightarrow\mathrm{x}=\mathrm{3} \\ $$$$\mathrm{3}^{\mathrm{2}} −\mathrm{1}=\mathrm{2}^{\mathrm{3}} \checkmark \\ $$$$\mathrm{x}=\mathrm{3} \\ $$

Commented by MathsFan last updated on 07/Jun/22

thanks a lot :)

$$\left.{thanks}\:{a}\:{lot}\::\right) \\ $$

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