Question

the arihmatetic mean of two numbers 6. if the geometric mean is also 6, then find the numbers

Answer 1

Step-by-step explanation:

$\frac{ \alpha + \beta }{2} = \frac{6}{1} \\ \\ \alpha \beta = 6 \\ \\ \alpha + \beta = 12 \\ \\ \alpha + \frac{6}{ \alpha } = 12 \\ \\ { \alpha }^{2} - 12 \alpha + 6 = 0 \\ \\ \alpha = \frac{12 + \sqrt{144 - 24} }{2 } = \frac{12 + \sqrt{120} }{2} = \frac{12 + \sqrt{4 \times 30} }{2} = \frac{12 + 2 \sqrt{30} }{2} = 6 + \sqrt{30} \\ \\ \beta = 6 - \sqrt{30}$

$numbers \: are \: \: 6 + \sqrt{30} \: \: and \: \: \: \: 6 - \sqrt{30} .$