**Understanding the Question:**

The question states that p:q is the sub-duplicate ratio of p-x^2 : q-x^2. We need to find the value of x^2.

**Sub-Duplicate Ratio:**

The sub-duplicate ratio of two quantities a and b is the ratio of the square root of a to the square root of b. Mathematically, it can be expressed as:

Sub-duplicate ratio = √a / √b

**Formulation of the Problem:**

According to the given question, p:q is the sub-duplicate ratio of p-x^2 : q-x^2. Let's formulate this statement mathematically:

√(p-x^2) / √(q-x^2) = p / q

**Squaring Both Sides:**

To solve the equation, let's square both sides:

[√(p-x^2) / √(q-x^2)]^2 = (p / q)^2

(p-x^2) / (q-x^2) = p^2 / q^2

**Cross-Multiplication:**

Next, let's cross-multiply the equation:

p^2(q-x^2) = q^2(p-x^2)

p^2q - p^2x^2 = q^2p - q^2x^2

**Rearranging the Equation:**

Rearranging the equation, we get:

p^2q - q^2p = p^2x^2 - q^2x^2

pq(p - q) = x^2(p^2 - q^2)

**Simplifying the Equation:**

Further simplifying the equation, we have:

pq(p - q) = x^2(p - q)(p + q)

**Dividing Both Sides:**

Now, let's divide both sides of the equation by (p - q):

pq = x^2(p + q)

**Simplifying Further:**

Finally, we can express x^2 in terms of p and q:

x^2 = (pq) / (p + q)

Therefore, x^2 is equal to the ratio of the product of p and q to the sum of p and q.

**Conclusion:**

In conclusion, x^2 is equal to (pq) / (p + q) based on the given sub-duplicate ratio.