Solve (1/x + x)²: Expanding a Perfect Square Expression

$(\frac{1}{x}+x)^2=$

To solve this problem, we'll use the formula for the square of a sum.

Let's define our terms:
Let $a = \frac{1}{x}$ and $b = x$.

According to the formula $(a + b)^2 = a^2 + 2ab + b^2$, we need to find the following:
1. $a^2 = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2}$
2. $2ab = 2 \times \frac{1}{x} \times x = 2$
3. $b^2 = x^2$

Substituting these into the formula gives us:

$(a + b)^2 = \frac{1}{x^2} + 2 + x^2$

To combine these into a single fraction, find a common denominator, which is $x^2$:

• Convert $2$ to a fraction with $x^2$ as the denominator: $2 = \frac{2x^2}{x^2}$
• Convert $x^2$ to a fraction with $x^2$ as the denominator: $x^2 = \frac{x^4}{x^2}$

So, the expression becomes:

$\frac{1}{x^2} + \frac{2x^2}{x^2} + \frac{x^4}{x^2} = \frac{1 + 2x^2 + x^4}{x^2}$

Therefore, the expanded expression is $\frac{x^4 + 2x^2 + 1}{x^2}$.