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

The objective is to determine the value of \( x^2 + \frac{1}{x^2} \), given the equation \( x + \frac{1}{x} = 3 \).

1. Key Concepts:

- Algebraic Identity Utilized: The relevant identity is: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] - This identity establishes a relationship between the square of a binomial and the sum of squares.

2. Provided Information:

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

3. Calculation Steps:

Applying the identity: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] Substituting the known value: \[ 3^2 = x^2 + \frac{1}{x^2} + 2 \Rightarrow 9 = x^2 + \frac{1}{x^2} + 2 \] Isolating the target expression: \[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

Final Result:

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