Find the quadratic equation, if one of the roots is `sqrt(5) - sqrt(3)` and other is `sqrt(5)+sqrt(3)`.

Answer

To find the quadratic equation given the roots \( \sqrt{5} - \sqrt{3} \) and \( \sqrt{5} + \sqrt{3} \), we can follow these steps:

### Step 1: Identify the roots
Let the roots be:
- \( \alpha = \sqrt{5} - \sqrt{3} \)
- \( \beta = \sqrt{5} + \sqrt{3} \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = (\sqrt{5} - \sqrt{3}) + (\sqrt{5} + \sqrt{3}) = 2\sqrt{5} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated using the difference of squares:
\[ \alpha \beta = (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

### Step 4: Form the quadratic equation
Using the sum and product of the roots, we can form the quadratic equation using the standard form:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we calculated:
\[ x^2 - (2\sqrt{5})x + 2 = 0 \]

### Final Quadratic Equation
Thus, the quadratic equation is:
\[ x^2 - 2\sqrt{5}x + 2 = 0 \]

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