We are asked to find the roots of the equation $x^2 + \frac{13}{2}x = \frac{7}{2}x$.

1. Problem Description

We are asked to find the roots of the equation $x^2 + \frac{13}{2}x = \frac{7}{2}x$.

2. Solution Steps

First, we subtract $\frac{7}{2}x$ from both sides of the equation:

$x^2 + \frac{13}{2}x - \frac{7}{2}x = 0$

Combine the $x$ terms:

$x^2 + (\frac{13}{2} - \frac{7}{2})x = 0$

$x^2 + \frac{6}{2}x = 0$

$x^2 + 3x = 0$

Now, we can factor out an $x$:

$x(x + 3) = 0$

To find the roots, we set each factor equal to zero:

$x = 0$ or $x + 3 = 0$

So, $x = 0$ or $x = -3$.

We can check these roots in the original equation:

For $x = 0$:

$0^2 + \frac{13}{2}(0) = \frac{7}{2}(0)$

$0 + 0 = 0$

$0 = 0$, so $x = 0$ is a root.

For $x = -3$:

$(-3)^2 + \frac{13}{2}(-3) = \frac{7}{2}(-3)$

$9 - \frac{39}{2} = -\frac{21}{2}$

$\frac{18}{2} - \frac{39}{2} = -\frac{21}{2}$

$-\frac{21}{2} = -\frac{21}{2}$, so $x = -3$ is a root.

3. Final Answer

$x = 0, -3$