If arithmetic mean and geometric mean of two positive numbers a and b are 10 and 8 respectively.find the numbers

Given that the arithmetic mean (AM) and geometric mean (GM) of two positive numbers \( a \) and \( b \) are 10 and 8, respectively, we need to find the values of \( a \) and \( b \).

### Step 1: Set up the equations
The arithmetic mean of \( a \) and \( b \) is given by:

\[
\text{AM} = \frac{a + b}{2} = 10
\]

The geometric mean of \( a \) and \( b \) is given by:

\[
\text{GM} = \sqrt{ab} = 8
\]

### Step 2: Solve the AM equation
From the AM equation:

\[
\frac{a + b}{2} = 10
\]

Multiply both sides by 2:

\[
a + b = 20
\]

### Step 3: Solve the GM equation
From the GM equation:

\[
\sqrt{ab} = 8
\]

Square both sides to eliminate the square root:

\[
ab = 64
\]

### Step 4: Solve the system of equations
We now have the system of equations:

\[
a + b = 20 \quad \text{(1)}
\]
\[
ab = 64 \quad \text{(2)}
\]

To find \( a \) and \( b \), we solve this system. 

### Step 5: Use the quadratic formula
Consider the quadratic equation whose roots are \( a \) and \( b \):

\[
x^2 - (a + b)x + ab = 0
\]

Substituting from equations (1) and (2):

\[
x^2 - 20x + 64 = 0
\]

### Step 6: Solve the quadratic equation
We can solve this using the quadratic formula:

\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(64)}}{2(1)}
\]

\[
x = \frac{20 \pm \sqrt{400 - 256}}{2}
\]

\[
x = \frac{20 \pm \sqrt{144}}{2}
\]

\[
x = \frac{20 \pm 12}{2}
\]

So, the two possible values for \( x \) are:

\[
x = \frac{20 + 12}{2} = \frac{32}{2} = 16
\]
\[
x = \frac{20 - 12}{2} = \frac{8}{2} = 4
\]

### Final Answer:
The two numbers \( a \) and \( b \) are 16 and 4.

Would you like further details or have any questions?

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Here are 5 related questions to further your understanding:
1. How do arithmetic and geometric means relate in inequalities?
2. Can you find the harmonic mean of the numbers 16 and 4?
3. What happens if \( a \) and \( b \) are negative?
4. How would you solve the problem if only the arithmetic mean was given?
5. Can you solve for \( a \) and \( b \) if their harmonic mean is given?

**Tip:** When dealing with quadratic equations, always check if the discriminant is a perfect square to ensure the solutions are real and rational.

Learn how to find two positive numbers given their arithmetic mean of 10 and geometric mean of 8. Detailed solution using arithmetic and geometric mean formulas, with step-by-step calculations.

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