To find the solution of the equation

$z = \frac{1}{2}(p^2 + q^2) + (p - x)(q - y)$

that passes through the x-axis, we need to set $z = 0$ because the x-axis corresponds to the points where the value of $z$ is zero.

So, we set the equation to zero:

$0 = \frac{1}{2}(p^2 + q^2) + (p - x)(q - y)$

Now, we can rearrange this equation to isolate terms involving $p$ and $q$:

$(p - x)(q - y) = -\frac{1}{2}(p^2 + q^2)$

To solve for $p$ and $q$, we can analyze this equation further, but typically it requires specific values for $x$ and $y$ or additional constraints. The relationship shows how $p$ and $q$ depend on $x$ and $y$.

If you have specific values for $x$ and $y$, we can substitute those in and solve for $p$ and $q$. Otherwise, you can express the relationship between these variables parametrically or graphically.

Would you like to provide specific values for $x$ and $y$ or further clarify the problem? Here are some related questions to consider:

1. What values do you want to assign to $x$ and $y$?
2. Would you like to explore the geometric interpretation of this equation?
3. Are you interested in finding the specific $p$ and $q$ values numerically or algebraically?
4. Should we consider any particular constraints or conditions on $p$ and $q$?
5. Would you like to visualize the relationship between $p$, $q$, $x$, and $y$?

Tip: When working with equations involving multiple variables, isolating one variable can often simplify the process of finding solutions.