Question

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

Answer 1

Answer:

4 and 16

Step-by-step explanation:

Let a and b be two unknown numbers.

1. The arithmetic mean of these numbers is

$\dfrac{a+b}{2}=10$

2. The geometric mean of these numbers is

$\sqrt{ab}=8$

Solve the system of two equations:

$\left\{\begin{array}{l}\dfrac{a+b}{2}=10\\ \\\sqrt{ab}=8\end{array}\right.\Rightarrow \left\{\begin{array}{l}a+b=20\\ \\ab=64\end{array}\right.$

From the first equation

$a=20-b$

Substitute it into the second equation

$(20-b)b=64\\ \\20b-b^2=64\\ \\-b^2+20b-64=0\\ \\b^2-20b+64=0\\ \\D=(-20)^2-4\cdot 64=400-256=144\\ \\b_{1,2}=\dfrac{-(-20)\pm \sqrt{144}}{2}=\dfrac{20\pm 12}{2}=16,\ 4$

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

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