The problem asks us to solve the quadratic equation $\frac{x^2}{2} + \frac{7}{2}x = 1$ by completing the square.

1. Problem Description

The problem asks us to solve the quadratic equation $\frac{x^2}{2} + \frac{7}{2}x = 1$ by completing the square.

2. Solution Steps

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

$x^2 + 7x = 2$

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, $b = 7$, so $(\frac{b}{2})^2 = (\frac{7}{2})^2 = \frac{49}{4}$.

Add $\frac{49}{4}$ to both sides:

$x^2 + 7x + \frac{49}{4} = 2 + \frac{49}{4}$

Rewrite the left side as a perfect square:

$(x + \frac{7}{2})^2 = 2 + \frac{49}{4}$

Convert 2 to a fraction with a denominator of 4: $2 = \frac{8}{4}$. So we have:

$(x + \frac{7}{2})^2 = \frac{8}{4} + \frac{49}{4} = \frac{57}{4}$

Take the square root of both sides:

$x + \frac{7}{2} = \pm \sqrt{\frac{57}{4}}$

$x + \frac{7}{2} = \pm \frac{\sqrt{57}}{2}$

Isolate $x$:

$x = -\frac{7}{2} \pm \frac{\sqrt{57}}{2}$

Combine the fractions:

$x = \frac{-7 \pm \sqrt{57}}{2}$

3. Final Answer

$x = \frac{-7 + \sqrt{57}}{2}, \frac{-7 - \sqrt{57}}{2}$