If root of a quadratic equation are `(3-sqrt7),(3+sqrt7)` and then find the quadratic equation

Answer

To find the quadratic equation given the roots \( (3 - \sqrt{7}) \) and \( (3 + \sqrt{7}) \), we can follow these steps:

### Step 1: Identify the Roots
Let the roots be:
- \( \alpha = 3 - \sqrt{7} \)
- \( \beta = 3 + \sqrt{7} \)

### Step 2: Calculate the Sum of the Roots
The sum of the roots \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = (3 - \sqrt{7}) + (3 + \sqrt{7}) = 3 + 3 - \sqrt{7} + \sqrt{7} = 6 \]

### Step 3: Calculate the Product of the Roots
The product of the roots \( \alpha \cdot \beta \) can be calculated using the difference of squares formula:
\[ \alpha \cdot \beta = (3 - \sqrt{7})(3 + \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2 \]

### Step 4: Form the Quadratic Equation
The standard form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we calculated:
\[ x^2 - (6)x + (2) = 0 \]
Thus, the quadratic equation is:
\[ x^2 - 6x + 2 = 0 \]

### Final Answer
The quadratic equation is:
\[ \boxed{x^2 - 6x + 2 = 0} \]