Question

Find two numbers whose arithmetic mean is 10 and geometric mean is 8​

Answer 1

4 and 16

Explanation:

Let a and b be two unknown numbers.

1. The arithmetic mean of these numbers is

`(a+b)/(2)=10`

2. The geometric mean of these numbers is

`√(ab)=8`

Solve the system of two equations:

`\left\{\begin{array}{l}(a+b)/(2)=10\\ \\√(ab)=8\end{array}\right.\Rightarrow \left\{\begin{array}{l}a+b=20\\ \\ab=64\end{array}\right.`

From the first equation

`a=20-b`

Substitute it into the second equation

`(20-b)b=64\\ \\20b-b^2=64\\ \\-b^2+20b-64=0\\ \\b^2-20b+64=0\\ \\D=(-20)^2-4\cdot 64=400-256=144\\ \\b_(1,2)=(-(-20)\pm √(144))/(2)=(20\pm 12)/(2)=16,\ 4`

When
`b=16,\ a=20-b=20-16=4`

When
`b=4,\ a=20-b=20-4=16`