Solve the equation $\frac{x^2}{2} + \frac{5}{2}x = 2$ by completing the square.

1. Problem Description

Solve the equation $\frac{x^2}{2} + \frac{5}{2}x = 2$ by completing the square.

2. Solution Steps

First, multiply both sides of the equation by 2 to eliminate the fractions:

$x^2 + 5x = 4$

To complete the square, we need to add $(\frac{b}{2})^2$ to both sides of the equation, where $b$ is the coefficient of the $x$ term. In this case, $b=5$.

So, we add $(\frac{5}{2})^2 = \frac{25}{4}$ to both sides:

$x^2 + 5x + \frac{25}{4} = 4 + \frac{25}{4}$

Now, the left side is a perfect square:

$(x + \frac{5}{2})^2 = 4 + \frac{25}{4}$

Simplify the right side:

$(x + \frac{5}{2})^2 = \frac{16}{4} + \frac{25}{4} = \frac{41}{4}$

Take the square root of both sides:

$x + \frac{5}{2} = \pm \sqrt{\frac{41}{4}} = \pm \frac{\sqrt{41}}{2}$

Isolate $x$:

$x = -\frac{5}{2} \pm \frac{\sqrt{41}}{2}$

Combine the fractions:

$x = \frac{-5 \pm \sqrt{41}}{2}$

The two solutions are $x = \frac{-5 + \sqrt{41}}{2}$ and $x = \frac{-5 - \sqrt{41}}{2}$.

3. Final Answer

$\frac{-5 + \sqrt{41}}{2}, \frac{-5 - \sqrt{41}}{2}$