If one root of a quadratic equation is `2 + sqrt5`, then find the quadratic equation.

Answer

To find the quadratic equation given one root as \(2 + \sqrt{5}\), we can follow these steps:

### Step 1: Identify the Roots
Given one root of the quadratic equation is \( \alpha = 2 + \sqrt{5} \). Since the coefficients of a quadratic equation are real numbers, the other root \( \beta \) must be the conjugate of \( \alpha \):
\[ \beta = 2 - \sqrt{5} \]

### Step 2: Calculate the Sum of the Roots
The sum of the roots \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = (2 + \sqrt{5}) + (2 - \sqrt{5}) = 2 + 2 = 4 \]

### Step 3: Calculate the Product of the Roots
The product of the roots \( \alpha \cdot \beta \) is calculated as:
\[ \alpha \cdot \beta = (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \]

### 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 can substitute the values we found:
\[ x^2 - (4)x + (-1) = 0 \]
This simplifies to:
\[ x^2 - 4x - 1 = 0 \]

### Final Answer
The quadratic equation is:
\[ \boxed{x^2 - 4x - 1 = 0} \]