To solve the problem where the sum and difference of the arithmetic mean (AM) and geometric mean (GM) between two positive numbers are given as 80 and 20 respectively, follow these steps:
1. Define the arithmetic mean and geometric mean:
- Arithmetic Mean (AM): [tex]\(\text{AM} = \frac{a + b}{2}\)[/tex]
- Geometric Mean (GM): [tex]\(\text{GM} = \sqrt{a \cdot b}\)[/tex]
2. Given information:
- The sum of AM and GM: [tex]\(\text{AM} + \text{GM} = 80\)[/tex]
- The difference of AM and GM: [tex]\(\text{AM} - \text{GM} = 20\)[/tex]
3. Solve for AM and GM:
- Add the two equations:
[tex]\[
(\text{AM} + \text{GM}) + (\text{AM} - \text{GM}) = 80 + 20
\][/tex]
[tex]\[
2\text{AM} = 100 \implies \text{AM} = 50
\][/tex]
- Subtract the second equation from the first:
[tex]\[
(\text{AM} + \text{GM}) - (\text{AM} - \text{GM}) = 80 - 20
\][/tex]
[tex]\[
2\text{GM} = 60 \implies \text{GM} = 30
\][/tex]
4. Set up the equations using AM and GM values:
- Since [tex]\(\text{AM} = \frac{a + b}{2} = 50\)[/tex], we get:
[tex]\[
a + b = 100
\][/tex]
- Since [tex]\(\text{GM} = \sqrt{a \cdot b} = 30\)[/tex], we get:
[tex]\[
a \cdot b = 900
\][/tex]
5. Form a quadratic equation using [tex]\(a + b\)[/tex] and [tex]\(a \cdot b\)[/tex]:
[tex]\[
x^2 - (a + b)x + a \cdot b = 0
\][/tex]
- Substitute the known values:
[tex]\[
x^2 - 100x + 900 = 0
\][/tex]
6. Solve for the roots of the quadratic equation:
- Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
[tex]\[
x = \frac{100 \pm \sqrt{100^2 - 4 \cdot 900}}{2}
\][/tex]
[tex]\[
x = \frac{100 \pm \sqrt{10000 - 3600}}{2}
\][/tex]
[tex]\[
x = \frac{100 \pm \sqrt{6400}}{2}
\][/tex]
[tex]\[
x = \frac{100 \pm 80}{2}
\][/tex]
- Calculate the two roots:
[tex]\[
x1 = \frac{100 + 80}{2} = 90
\][/tex]
[tex]\[
x2 = \frac{100 - 80}{2} = 10
\][/tex]
The two positive numbers are 90 and 10.
