question_answer

If \[x+\frac{1}{x}=5,\], find the value of \[{{x}^{2}}+\frac{1}{{{x}^{2}}}.\]

Answer:

Since, \[x+\frac{1}{x}=5\]

Squaring on both sides,

\[{{\left( x+\frac{1}{x} \right)}^{2}}={{5}^{2}}\]

or \[{{x}^{2}}+\frac{1}{{{x}^{2}}}+2\times x\times \frac{1}{x}={{(5)}^{2}}\]

or  \[{{x}^{2}}+\frac{1}{{{x}^{2}}}+2=25\]

or \[{{x}^{2}}+\frac{1}{{{x}^{2}}}=25-2\]

or \[{{x}^{2}}+\frac{1}{{{x}^{2}}}=23\]