If p,q and r rational numbers, then 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. ` 7-4sqrt(3)`
D. 4

Explanation

To solve the quadratic equation \( (2 - \sqrt{3})x^2 - (7 - 4\sqrt{3})x + (2 + \sqrt{3}) = 0 \), we can use the quadratic formula, which states that for any quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

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

### Step 2: Calculate the discriminant
Next, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]

Calculating \( b^2 \):
\[ b^2 = (-7 + 4\sqrt{3})^2 = 49 - 56\sqrt{3} + 48 = 97 - 56\sqrt{3} \]

Calculating \( 4ac \):
\[ 4ac = 4(2 - \sqrt{3})(2 + \sqrt{3}) = 4((2)^2 - (\sqrt{3})^2) = 4(4 - 3) = 4 \]

Now substituting these values into the discriminant:
\[ D = (97 - 56\sqrt{3}) - 4 = 93 - 56\sqrt{3} \]

### Step 3: Calculate the roots using the quadratic formula
Now we can substitute \( a \), \( b \), and \( D \) into the quadratic formula:
\[ x = \frac{-(-7 + 4\sqrt{3}) \pm \sqrt{93 - 56\sqrt{3}}}{2(2 - \sqrt{3})} \]
\[ x = \frac{7 - 4\sqrt{3} \pm \sqrt{93 - 56\sqrt{3}}}{4 - 2\sqrt{3}} \]

### Step 4: Simplify the expression
To simplify further, we can rationalize the denominator if necessary, but for now, we have the roots expressed in terms of the square root.

### Final Result
The roots of the equation \( (2 - \sqrt{3})x^2 - (7 - 4\sqrt{3})x + (2 + \sqrt{3}) = 0 \) are:
\[ x = \frac{7 - 4\sqrt{3} \pm \sqrt{93 - 56\sqrt{3}}}{4 - 2\sqrt{3}} \]