Construct a quadratic equation in x such that the A.M. of its roots is A and G.M. is G.

Answer

To construct a quadratic equation in \( x \) such that the Arithmetic Mean (A.M.) of its roots is \( A \) and the Geometric Mean (G.M.) is \( G \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \).

2. **Using the A.M.**:
The A.M. of the roots \( \alpha \) and \( \beta \) is given by:
\[ \text{A.M.} = \frac{\alpha + \beta}{2} = A \]
From this, we can derive:
\[ \alpha + \beta = 2A \quad \text{(Equation 1)} \]

3. **Using the G.M.**:
The G.M. of the roots \( \alpha \) and \( \beta \) is given by:
\[ \text{G.M.} = \sqrt{\alpha \beta} = G \]
Squaring both sides, we get:
\[ \alpha \beta = G^2 \quad \text{(Equation 2)} \]

4. **Forming the Quadratic Equation**:
A quadratic equation can be expressed in the form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values from Equations 1 and 2:
\[ x^2 - (2A)x + G^2 = 0 \]

5. **Final Quadratic Equation**:
Therefore, the quadratic equation we are looking for is:
\[ x^2 - 2Ax + G^2 = 0 \]

### Summary:
The quadratic equation in \( x \) such that the A.M. of its roots is \( A \) and the G.M. is \( G \) is:
\[ x^2 - 2Ax + G^2 = 0 \]