Let and be the roots of the quadratic equation x^2-2bx+c^2=0 where b , c are positive real numbers. Let A be the arithmetic mean of and ; and G be the geometric mean of and . What are the roots of the quadratic equation x^2-(b+c)x+bc=0 ?

Step-by-step solution

Given the quadratic equation x^2 - 2bx + c^2 = 0 with roots and . Sum of the roots is + = 2b . Product of the roots is = c^2 . The arithmetic mean A of and is given by A = + 2 = 2b 2 = b . The geometric mean G of and is given by G = = c^2 = c (since c is a positive real number). The second quadratic equation is x^2 - (b+c)x + bc = 0 . Factoring the equation, we get (x - b)(x - c) = 0 . The roots are x = b and x = c . Substituting b = A and c = G , the roots are A and G . Answer: A, G