The value of x2 + 1/x2 for x + 1/x = 5 is

Concept

For any nonzero real number x, if x + 1/x = k for a constant k, squaring both sides gives (x + 1/x)2 = k2. Expanding the left side with (a + b)2 = a2 + 2ab + b2, and using x·(1/x) = 1, produces x2 + 2 + 1/x2 = k2. This gives the general identity x2 + 1/x2 = k2 − 2, linking a sum-reciprocal expression to a sum-of-squares expression.

Application

1. Start from the given relation: x + 1/x = 5.

2. Square both sides: (x + 1/x)2 = 52 = 25.

3. Expand the left side using (a + b)2 = a2 + 2ab + b2: x2 + 2·x·(1/x) + 1/x2 = 25.

4. Simplify the middle term, since x·(1/x) = 1 for every nonzero x: x2 + 2 + 1/x2 = 25.

5. Subtract 2 from both sides to isolate the required expression: x2 + 1/x2 = 25 − 2 = 23.

Cross-check

Using the general identity derived above with k = 5: x2 + 1/x2 = k2 − 2 = 52 − 2 = 25 − 2 = 23, which matches the step-by-step expansion and confirms the value is consistent with the given sum.