If p:q is the sub- duplicate ratio of p-x 2:q-x2 then X2 is.?

Question

Answer

Explanation:

Let's assume that the sub-duplicate ratio of p:q is k. So we can write:

p:k = k:q

By cross-multiplication, we get:

pq = k²

Now, let's assume that p-x and q-x are the terms of the sub-duplicate ratio. So we can write:

p-x:k-x = k:p-x-q

By cross-multiplication, we get:

(p-x)(p-x-q) = k(k-x)

Expanding the brackets, we get:

p² - px - pq + x² = k² - kx

Substituting pq with k², we get:

p² - px - k² + x² = -kx

Moving all the terms to one side, we get:

x² - (p + q)x + (p² - k²) = 0

This is a quadratic equation in x. We can use the quadratic formula to find x:

x = [(p + q) ± √((p + q)² - 4(p² - k²))]/2

x = [(p + q) ± √(p² + 2pq + q² - 4p² + 4k²)]/2

x = [(p + q) ± √(q² - 2pq + p² + 4k²)]/2

x = [(p + q) ± √((q - p)² + 4k²)]/2

x² = [(p + q) ± √((q - p)² + 4k²)]/2

Therefore, x² is:

x² = [(p + q) ± √((q - p)² + 4k²)]/2

Answer

Answer= pq/p+q = x²