Explanation:

To solve the problem, we need to use the concept of sub-duplicate ratios.

Definition of Sub-duplicate Ratio:

The sub-duplicate ratio is the ratio of the square roots of two quantities. If the ratio of two quantities p and q is given by p:q, then the sub-duplicate ratio of p:q is √p:√q.

Given:

- The ratio of p and q is p:q.
- The ratio of p-x² and q-x² is also p:q.

Steps to Solve:

We can use the concept of sub-duplicate ratio to solve for x² as follows:

1. Write the sub-duplicate ratio of p:q as √p:√q.

2. Write the sub-duplicate ratio of p-x²:q-x² as √(p-x²):√(q-x²).

3. Since the ratio of p and q is p:q, we can write p/q as a constant k.

4. Using the above equation, we can write p = kq.

5. Similarly, using the given ratio of p-x² and q-x², we can write (p-x²)/(q-x²) = k.

6. Cross-multiplying and simplifying the above equation, we get p-q = kx².

7. Substituting the value of p = kq from step 4, we get kq-q = kx².

8. Simplifying the above equation, we get x² = q/(k+1).

9. Substituting the value of k = p/q from step 3, we get x² = q/(p/q+1).

10. Simplifying the above equation, we get x² = q/(p+q).

Therefore, x² = q/(p+q).

Conclusion:

The value of x² is q/(p+q), which can be obtained using the sub-duplicate ratio concept and the given ratios of p:q and p-x²:q-x².