The real quadratic equation whose one root is `2-sqrt(3)` is

Options

A. `x^(2)-4x+1=0`
B. `x^(2)+4x-1=0`
C. `x^(2)-4x-1=0`
D. none of these

Answer

To find the real quadratic equation whose one root is \(2 - \sqrt{3}\), we can follow these steps:

### Step 1: Identify the roots
Given one root \( \alpha = 2 - \sqrt{3} \), we can find the second root \( \beta \). For a quadratic equation with real coefficients, if one root is \( a - b \), the other root is typically \( a + b \). Therefore, the second root is:
\[ \beta = 2 + \sqrt{3} \]

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) is calculated as follows:
\[ \alpha + \beta = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4 \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \cdot \beta \) can be calculated using the difference of squares:
\[ \alpha \cdot \beta = (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 4: Form the quadratic equation
Using the standard form of a quadratic equation \( x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \), we substitute the values we found:
\[ x^2 - (4)x + (1) = 0 \]
This simplifies to:
\[ x^2 - 4x + 1 = 0 \]

### Final Answer
The real quadratic equation whose one root is \(2 - \sqrt{3}\) is:
\[ \boxed{x^2 - 4x + 1 = 0} \]