If \(X + \frac{1}{X} = 15\), find the value of \(X^2 + \frac{1}{X^2}\).
Solving for \(X^2 + \frac{1}{X^2}\)
We are given the equation:
\( X + \frac{1}{X} = 15 \)
Our goal is to find the value of \(X^2 + \frac{1}{X^2}\).
Algebraic Manipulation
To find \(X^2 + \frac{1}{X^2}\), we can square both sides of the given equation.
- Square both sides: \((X + \frac{1}{X})^2 = 15^2\)
- Expand the left side using the identity \((a+b)^2 = a^2 + 2ab + b^2\): \(X^2 + 2 \cdot X \cdot \frac{1}{X} + \left(\frac{1}{X}\right)^2 = 225\)
- Simplify the middle term and the squared term: \(X^2 + 2 + \frac{1}{X^2} = 225\)
- Isolate the term \(X^2 + \frac{1}{X^2}\) by subtracting 2 from both sides: \(X^2 + \frac{1}{X^2} = 225 - 2\)
- Calculate the final value: \(X^2 + \frac{1}{X^2} = 223\)
Therefore, the value of \(X^2 + \frac{1}{X^2}\) is 223.
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