the AM between two positive numbers x and y is 3 and G.M is square root of 7 then x square + y square

Given the problem:

- The arithmetic mean (AM) between two positive numbers \( x \) and \( y \) is 3.
- The geometric mean (GM) between the same numbers is \( \sqrt{7} \).

We need to find \( x^2 + y^2 \).

### Step 1: Use the Arithmetic Mean formula
The arithmetic mean of \( x \) and \( y \) is given by:

\[
\text{AM} = \frac{x + y}{2} = 3
\]

Solving for \( x + y \):

\[
x + y = 6 \quad \text{(Equation 1)}
\]

### Step 2: Use the Geometric Mean formula
The geometric mean of \( x \) and \( y \) is given by:

\[
\text{GM} = \sqrt{xy} = \sqrt{7}
\]

Squaring both sides:

\[
xy = 7 \quad \text{(Equation 2)}
\]

### Step 3: Find \( x^2 + y^2 \)
We know the identity:

\[
x^2 + y^2 = (x + y)^2 - 2xy
\]

Substitute \( x + y = 6 \) and \( xy = 7 \):

\[
x^2 + y^2 = 6^2 - 2 \times 7
\]

Simplify the expression:

\[
x^2 + y^2 = 36 - 14 = 22
\]

So, \( x^2 + y^2 = \boxed{22} \).

Would you like more details on this solution or have any questions?

### Related Questions
1. What are the values of \( x \) and \( y \) individually?
2. How does the relationship between AM, GM, and HM (Harmonic Mean) work?
3. What happens if the AM is greater than the GM for any two numbers?
4. How can we derive the quadratic equation from the given AM and GM?
5. What is the minimum possible value of \( x^2 + y^2 \) given these conditions?

### Tip
The relationship between AM, GM, and HM is a useful tool in solving problems involving pairs of numbers, especially when these means are given.

Learn how to find x^2 + y^2 when given the arithmetic mean (AM) and geometric mean (GM) of two positive numbers. Follow a detailed step-by-step solution with key mathematical formulas and identities used. Suitable for students in Grades 9-12.

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