Introduction:
In this problem, we are given that p:q is sub-duplicate ratio of p-x² : q-x². We need to find x². Let us solve this problem step by step.

Solution:
To solve this problem, we will use the concept of sub-duplicate ratio. Sub-duplicate ratio is a ratio in which the square of the first term is less than the product of the first and second terms, and the square of the second term is greater than the product of the first and second terms. In other words,



Step 1: Write the sub-duplicate ratio of p-x² : q-x²



Step 2: Apply the concept of sub-duplicate ratio



Step 3: Simplify the above expressions



Step 4: Simplify further



Step 5: Combine like terms



Step 6: Factor the expression



where α and β are the roots of the quadratic equation p² - 2pq + q² = 0.

Step 7: Find the values of α and β



Step 8: Find the range of x²



Step 9: Substitute the values of p and q



Step 10: Substitute the given ratio p