Let and be the roots of the quadratic equation x^2 - px + q = 0 , where p = 5^ ₅ k - k^ _k 5 + 7 (for k > 1, k 5 ) and q = 9^ _ 3 3 8 - m . If the quadratic equation whose roots are + and is of the form x^2 - 11x + c = 0 , then the value of the integer m is

Step-by-step solution

Given the quadratic equation x^2 - px + q = 0 with roots and . Using the identity a^ _a b = b^ _b a , we can simplify p : p = 5^ ₅ k - k^ _k 5 + 7 = 0 + 7 = 7 . By Vieta's formulas, the sum of the roots is + = p = 7 and the product of the roots is = q . The new quadratic equation has roots + and , which are 7 and q . The sum of these new roots is 7 + q . We are given that the new equation is of the form x^2 - 11x + c = 0 , so the sum of its roots is 11 . Therefore, 7 + q = 11 q = 4 . Now, we evaluate the logarithmi