The arithmetic mean of two numbers is 18 3 4 and the positive square root of their product is 15 . The larger of the two numbers is

Step-by-step solution

Let a and b be the positive numbers. Then a + b 2 = 18 3 4 ⇒ a + b = 75 2 and a b = 15 ⇒ a b = 225 so, a - b 2 = a + b 2 - 4 a b = 75 2 2 - 4 × 225 = 75 2 + 30 75 2 - 30 = 135 2 × 15 2 ⇒ a - b = ± 15 2 × 3 = ± 45 2 Case1 → a + b = 75 2 , a - b = 45 2 ⇒ a = 30 , b = 45 2 Case2 → a + b = 75 2 , a - b = - 45 2 ⇒ a = 15 2 , b = 30 So, larger of the two numbers is 30