Question

If \[x + \frac{1}{x} = 20,\]find the value of \[x^2 + \frac{1}{x^2} .\].

Answer 1

Let us consider the following equation: \[x + \frac{1}{x} = 20\]

Squaring both sides, we get:

\[\left( x + \frac{1}{x} \right)^2 = \left( 20 \right)^2 = 400\]

\[ \Rightarrow \left( x + \frac{1}{x} \right)^2 = 400\]

\[ \Rightarrow x^2 + 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 400 [(a + b )^2 = a^2 + b^2 + 2ab]\]

\[ \Rightarrow x^2 + 2 + \frac{1}{x^2} = 400\]

\[\Rightarrow x^2 + \frac{1}{x^2} = 398\]                    (Subtracting 2 from both sides)

Thus, the answer is 398.