If $x + \dfrac{1}{x} = 5$, find $x^{2} + \dfrac{1}{x^{2}}$.

If $x + \dfrac{1}{x} = 5$, find $x^{2} + \dfrac{1}{x^{2}}$.
  1. $23$
  2. $25$
  3. $27$
  4. $21$

Solution

$\left(x+\dfrac{1}{x}\right)^{2} = x^{2} + 2 + \dfrac{1}{x^{2}} = 25 \Rightarrow x^{2}+\dfrac{1}{x^{2}} = 23$.

Asked in: IMO

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