Understanding the Sub-Duplicate Ratio
The sub-duplicate ratio of two quantities is the ratio of their differences to their square roots. In this case, we are given that p : q is the sub-duplicate ratio of p - x^2 : q - x^2.

Applying the Sub-Duplicate Ratio Formula
To find the sub-duplicate ratio of two quantities, we use the formula: (p - q) : (sqrt(p) - sqrt(q)). In this case, the given ratio is: p : q = (p - x^2) : (q - x^2).

Expanding the Given Ratios
Expanding the given ratios, we get:
p : q = (p - x^2) : (q - x^2)
=> p/q = (p - x^2)/(q - x^2)

Using the Sub-Duplicate Ratio Formula
Now, we can apply the sub-duplicate ratio formula:
(p - q) : (sqrt(p) - sqrt(q)) = (p - x^2) : (sqrt(p) - sqrt(x^2))

Solving for x^2
By comparing the given ratio with the formula, we can see that x^2 is the square of the difference between p and q. Therefore, x^2 = (p - q)^2.

Conclusion
In conclusion, x^2 is equal to the square of the difference between p and q. It is important to understand the concept of sub-duplicate ratio and how to apply the formula to solve for x^2 in this scenario.