Let and be the roots of the equation x^2 - 8x + q = 0 , and let and be the roots of the equation x^2 - 72x + s = 0 . If , , , form a strictly increasing geometric progression of positive real numbers, then the value of s + q is _____.

Step-by-step solution

Let the four roots , , , in strictly increasing G.P. be a, ar, ar^2, ar^3 . Since the terms are positive and strictly increasing, we have a > 0 and r > 1 . From the first quadratic equation x^2 - 8x + q = 0 , the sum of the roots is: + = 8 a + ar = 8 a(1 + r) = 8 ... (1) From the second quadratic equation x^2 - 72x + s = 0 , the sum of the roots is: + = 72 ar^2 + ar^3 = 72 ar^2(1 + r) = 72 ... (2) Dividing equation (2) by equation (1): ar^2(1 + r) a(1 + r) = 72 8 r^2 = 9 Since r > 1 , we get r = 3 . Substitute r =