What is the sum of the roots of the equation `(x-sqrt2)(x^2-sqrt3x+pi)=0` ?

Options

A. -0.315
B. -0.318
C. 1.414
D. 3.15

Explanation

To find the sum of the roots of the equation \((x - \sqrt{2})(x^2 - \sqrt{3}x + \pi) = 0\), we can follow these steps:

### Step 1: Identify the Roots
The given equation is a product of two factors set to zero. This means that we can find the roots from each factor separately.

1. The first factor is \(x - \sqrt{2} = 0\).
- This gives us one root: \(x_1 = \sqrt{2}\).

2. The second factor is \(x^2 - \sqrt{3}x + \pi = 0\).
- We will need to use the quadratic formula to find the roots of this equation.

### Step 2: Use the Quadratic Formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \(x^2 - \sqrt{3}x + \pi = 0\):
- Here, \(a = 1\), \(b = -\sqrt{3}\), and \(c = \pi\).

### Step 3: Calculate the Discriminant
First, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = (-\sqrt{3})^2 - 4 \cdot 1 \cdot \pi = 3 - 4\pi \]

### Step 4: Find the Roots
Using the quadratic formula:
\[ x = \frac{-(-\sqrt{3}) \pm \sqrt{3 - 4\pi}}{2 \cdot 1} = \frac{\sqrt{3} \pm \sqrt{3 - 4\pi}}{2} \]
Let the roots from this factor be:
- \(x_2 = \frac{\sqrt{3} + \sqrt{3 - 4\pi}}{2}\)
- \(x_3 = \frac{\sqrt{3} - \sqrt{3 - 4\pi}}{2}\)

### Step 5: Sum of the Roots
The sum of the roots of the entire equation is given by:
\[ \text{Sum} = x_1 + x_2 + x_3 \]
Substituting the values:
\[ \text{Sum} = \sqrt{2} + \frac{\sqrt{3} + \sqrt{3 - 4\pi}}{2} + \frac{\sqrt{3} - \sqrt{3 - 4\pi}}{2} \]
Notice that the terms \(\sqrt{3 - 4\pi}\) will cancel out:
\[ \text{Sum} = \sqrt{2} + \frac{\sqrt{3} + \sqrt{3}}{2} = \sqrt{2} + \sqrt{3} \]

### Final Answer
Thus, the sum of the roots of the equation is:
\[ \sqrt{2} + \sqrt{3} \]