Question

If $ x + \frac{1}{x} = 3 $, what is the value of $ x^2 + \frac{1}{x^2} $?

• A. 7
• B. 9
• C. 8
• D. 6

Hint:

Use the identity \( \left(x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2 \) to find squares when expressions involve reciprocals.

Answer 1

The Correct Option is A

To solve the problem, we need to find the value of \( x^2 + \frac{1}{x^2} \), given that \( x + \frac{1}{x} = 3 \).

1. Understanding the Concepts:

- Algebraic Identity: We use the identity: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] - This identity allows us to relate the square of a sum to the sum of squares.

2. Given Values:

\[ x + \frac{1}{x} = 3 \]

3. Calculating the Required Expression:

Use the identity: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] Substitute the given value: \[ 3^2 = x^2 + \frac{1}{x^2} + 2 \Rightarrow 9 = x^2 + \frac{1}{x^2} + 2 \] \[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

Final Answer:

The value of \( x^2 + \frac{1}{x^2} \) is 7.