Given the. two roots .`sqrt3` and `-3sqrt3` how rnany quadratic equations can be formed?

Options

A. 2
B. 1
C. 0
D. none of these

Answer

To determine how many quadratic equations can be formed given the roots \( \sqrt{3} \) and \( -3\sqrt{3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Roots**:
We are given two roots:
- \( \alpha = \sqrt{3} \)
- \( \beta = -3\sqrt{3} \)

2. **Use the Quadratic Equation Formula**:
The standard form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

3. **Calculate the Sum of the Roots**:
Find \( \alpha + \beta \):
\[ \alpha + \beta = \sqrt{3} + (-3\sqrt{3}) = \sqrt{3} - 3\sqrt{3} = -2\sqrt{3} \]

4. **Calculate the Product of the Roots**:
Find \( \alpha \beta \):
\[ \alpha \beta = \sqrt{3} \times (-3\sqrt{3}) = -3 \times 3 = -9 \]

5. **Substitute into the Quadratic Formula**:
Substitute the values of the sum and product into the quadratic equation:
\[ x^2 - (-2\sqrt{3})x + (-9) = 0 \]
This simplifies to:
\[ x^2 + 2\sqrt{3}x - 9 = 0 \]

6. **Conclusion**:
Since we have only one set of roots, we can form only one quadratic equation from these roots.

### Final Answer:
Only one quadratic equation can be formed with the given roots \( \sqrt{3} \) and \( -3\sqrt{3} \).

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