A basket contains 72 apples of different weights. The average weight of the apples in the basket is (m + 4) kg. If 24 apples with an average weight of (n + 2) kg are removed, the average weight of the basket reduces by 2 kg. Additionally, if 8 more apples are removed with an average weight of (n - 4) kg, then the average weight of the remaining apples becomes (2m - 8) kg. If the values of (m - 2) and (n - 4) are the roots of a quadratic equation, find the quadratic equation.

The total weight of the 72 apples originally in the basket = 72 X (m + 4) ATP, 72 X (m + 4) - 24 X (n + 2) = (m + 4 - 2) X (72 - 24) 72m + 288 - 24n - 48 = (m + 2) X 48 72m - 24n + 240 = 48m + 96 24m - 24n + 144 = 0 m - n + 6 = 0 m = n - 6 --- (I) Again ATP, [72 X (m + 4) ] - [24 X (n + 2) + 8 X (n - 4) ] = (2m - 8) X (72 - 32) 72m + 288 - (24n + 48 + 8n - 32) = (2m - 8) X 40 72m - 32n + 272 = 80m - 320 -32n + 592 = 8m 8m + 32n = 592 m + 4n = 74 Substituting m = n - 6, n - 6 + 4n = 74 5n = 80 n = 16 So, m = 16 - 6 = 10 Now, Roots are = (m - 2), (n - 4) = 8, 12 Required quadratic equation = (x - 8) (x - 12) = x² - 20x + 96 = 0