What is the sum of the roots of the equation `(2 - sqrt(3)) x^(2) - (7 - 4 sqrt(3)) x + (2 + sqrt(3)) = 0` ?

Options

A. `2 - sqrt(3))`
B. `2 + sqrt(3))`
C. `7 - 4 sqrt(3))`
D. 4

Answer

To find the sum of the roots of the quadratic equation \((2 - \sqrt{3}) x^{2} - (7 - 4 \sqrt{3}) x + (2 + \sqrt{3}) = 0\), we can use the formula for the sum of the roots of a quadratic equation, which is given by:

\[ \text{Sum of the roots} = -\frac{b}{a} \]

where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).

### Step 1: Identify coefficients
From the given equation, we can identify:
- \(a = 2 - \sqrt{3}\)
- \(b = -(7 - 4\sqrt{3}) = -7 + 4\sqrt{3}\)

### Step 2: Calculate the sum of the roots
Using the formula for the sum of the roots:

\[ \text{Sum of the roots} = -\frac{b}{a} = -\frac{-7 + 4\sqrt{3}}{2 - \sqrt{3}} = \frac{7 - 4\sqrt{3}}{2 - \sqrt{3}} \]

### Step 3: Rationalize the denominator
To simplify \(\frac{7 - 4\sqrt{3}}{2 - \sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(2 + \sqrt{3}\):

\[ \text{Sum of the roots} = \frac{(7 - 4\sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

### Step 4: Calculate the denominator
The denominator simplifies as follows:

\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 5: Calculate the numerator
Now we calculate the numerator:

\[ (7 - 4\sqrt{3})(2 + \sqrt{3}) = 7 \cdot 2 + 7 \cdot \sqrt{3} - 4\sqrt{3} \cdot 2 - 4\sqrt{3} \cdot \sqrt{3} \]

Calculating each term:

- \(7 \cdot 2 = 14\)
- \(7 \cdot \sqrt{3} = 7\sqrt{3}\)
- \(-4\sqrt{3} \cdot 2 = -8\sqrt{3}\)
- \(-4\sqrt{3} \cdot \sqrt{3} = -12\)

Combining these gives:

\[ 14 + 7\sqrt{3} - 8\sqrt{3} - 12 = 14 - 12 - \sqrt{3} = 2 - \sqrt{3} \]

### Step 6: Final result
Thus, the sum of the roots is:

\[ \text{Sum of the roots} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3} \]

### Final Answer:
The sum of the roots of the equation is \(2 - \sqrt{3}\).

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