If the two roots of any quadratic equation are `sqrt3` and `sqrt3`, how many such equations are possible?

Options

A. 0
B. 1
C. 2
D. infinite

Answer

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

### Step 1: Understand the Roots
The roots of the quadratic equation are given as \( \alpha = \sqrt{3} \) and \( \beta = \sqrt{3} \). Since both roots are the same, we have a repeated root.

### Step 2: Use the Quadratic Formula
A quadratic equation can be expressed in the form:
\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]

### Step 3: Calculate the Sum of the Roots
The sum of the roots \( \alpha + \beta \) is:
\[ \sqrt{3} + \sqrt{3} = 2\sqrt{3} \]

### Step 4: Calculate the Product of the Roots
The product of the roots \( \alpha \cdot \beta \) is:
\[ \sqrt{3} \cdot \sqrt{3} = 3 \]

### Step 5: Form the Quadratic Equation
Substituting the sum and product of the roots into the quadratic equation formula, we get:
\[ x^2 - (2\sqrt{3})x + 3 = 0 \]

### Step 6: Conclusion on the Number of Equations
This equation can be simplified to:
\[ x^2 - 2\sqrt{3}x + 3 = 0 \]
Since the coefficients can be multiplied by any non-zero constant \( k \) (where \( k \) is a real number), we can form equations like:
\[ k(x^2 - 2\sqrt{3}x + 3) = 0 \]
However, all these equations represent the same set of roots. Therefore, there is only **one unique quadratic equation** with the roots \( \sqrt{3} \) and \( \sqrt{3} \).

### Final Answer
Thus, the number of such equations possible is **1**.

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