Find the value of x squared + 1/x squared if x + 1/x = 5

Question

If $x + \dfrac{1}{x} = 5$, find the value of $x^2 + \dfrac{1}{x^2}$.

Solution — Step by Step

Why This Works

This is a classic “reciprocal expression” problem. The trick is recognizing that squaring $x + \frac{1}{x}$ naturally produces $x^2 + \frac{1}{x^2}$ with an extra constant ($+2$) that we can subtract away.

The reason the middle term simplifies so cleanly is that $x \cdot \frac{1}{x} = 1$ regardless of the value of $x$. This “reciprocal property” makes these problems very tractable.

Alternative Method — Numerical Verification

We can find $x$ from $x + \frac{1}{x} = 5$: multiply by $x$ to get $x^2 - 5x + 1 = 0$.

$x = \frac{5 \pm \sqrt{21}}{2}$

Take $x = \frac{5 + \sqrt{21}}{2} \approx \frac{5 + 4.58}{2} \approx 4.79$

Then $x^2 \approx 22.94$ and $\frac{1}{x^2} \approx 0.044$.

$x^2 + \frac{1}{x^2} \approx 22.94 + 0.044 \approx 23$ ✓

The algebraic method is far faster than computing irrational roots.

Common Mistake