Given:
[tex]\sf x\ -\ \dfrac{1}{x}\ =\ 5[/tex]
To find:
[tex]\sf The\ value\ of\ x^2\ +\ \dfrac{1}{x^2}.[/tex]
Answer:
[tex]\sf x\ -\ \dfrac{1}{x}\ =\ 5[/tex]
Let's square the above expression.
[tex]\sf \bigg(x\ -\ \dfrac{1}{x}\bigg)^2\ =\ 5^2[/tex]
According to the identities, (a - b)² = a² - 2ab + b². Using the same logic for the expression above,
[tex]\sf x^2\ -\ 2\ \times\ x\ \times\ \dfrac{1}{x}\ +\ \dfrac{1}{x^2}\ =\ 25\\\\\\x^2\ -\ 2\ +\ \dfrac{1}{x^2}\ =\ 25\\\\\\x^2\ +\ \dfrac{1}{x^2}\ =\ 25 + 2\\\\\\x^2\ +\ \dfrac{1}{x^2}\ =\ 27[/tex]
Therefore, the value of x² + 1/x² is 27.
