Answer:
Since the arithmetic mean (average) is 5, (a + b)/2 =5 so a +b = 10 The geometric mean of two numbers is the square root of their product, so square root of ab = 3. So we have a system of equations where a +b = 10 and ab = 9. if we solve the first equation for b, we get b = 10 – a. Substituting this last expression for b into ab = 9, we arrive at a(10 – a) = 9. Distributing this last equation, we see that 10a – a squared = 9. Subtracting 9 from both sides and rearranging terms from highest power to lowest power, – a squared + 10a – 9 = 0. Next, multiplying this last equation by – 1, it becomes a squared – 10a + 9 = 0. Factoring, we see that (a – 1)(a – 9) = 0. Setting each factor equal to 0, either a – 1 =0, or a – 9 = 0. Thus, a = 1 or a = 9. When a = 1, b = 9 (from the equation b = 10 – a); when a = 9, b = 1 (also from the equation b = 10 – a). So the two numbers are 9 and 1.