Question

Find the mean proportional between `a - b and a^3 - a^2b`

Answer 1

Let the mean proportional between a – b and a³ – a²b be x.
⇒ a – b, x, a³ – a²b are in continued proportion.
⇒ a – b : x = x : a³ – a²b
⇒ x × x = (a – b) (a³ – a²b)
⇒ x² = (a – b) a²(a – b) = [a(a – b)]²
⇒ x = a(a – b)