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 ?

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Correct answer: A. A,\ G

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

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