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The equation you provided is:

\[
p(2x - p) + q(2x - q) = 2pq
\]

Let's simplify and solve for \(x\).

### Step 1: Expand both sides
Expand the left-hand side of the equation:

\[
p(2x - p) = 2px - p^2
\]
\[
q(2x - q) = 2qx - q^2
\]

Now, substituting these into the equation:

\[
(2px - p^2) + (2qx - q^2) = 2pq
\]

### Step 2: Combine like terms
Group the terms involving \(x\) on the left-hand side:

\[
2px + 2qx - p^2 - q^2 = 2pq
\]
Factor out the \(x\) terms:

\[
2x(p + q) - p^2 - q^2 = 2pq
\]

### Step 3: Isolate \(x\)
Move the constants to the right-hand side:

\[
2x(p + q) = 2pq + p^2 + q^2
\]

Now, solve for \(x\) by dividing both sides by \(2(p + q)\):

\[
x = \frac{2pq + p^2 + q^2}{2(p + q)}
\]

Thus, the solution for \(x\) is:

\[
x = \frac{2pq + p^2 + q^2}{2(p + q)}
\]

Would you like any more details or clarifications on the steps?

Here are five related questions to explore further:
1. What happens to \(x\) if \(p = q\)?
2. Can we simplify the equation further if \(p\) or \(q\) equals zero?
3. How does the equation change if we add a constant term to one side?
4. Can this equation represent any geometric shape when graphed?
5. What are the domain restrictions for \(x\) in terms of \(p\) and \(q\)?

**Tip:** When dealing with quadratic terms like \(p^2\) and \(q^2\), always group them together to simplify the equation before solving.

Solve the equation p(2x - p) + q(2x - q) = 2pq.

Learn how to solve the algebraic equation p(2x - p) + q(2x - q) = 2pq with step-by-step guidance. This problem focuses on algebraic expansion, factoring, and isolating the variable x. Suitable for high school students in Grades 9-11.

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