Two sides of a tariangle are given by the roots of the equation `x^(2) -2sqrt3 x+2 =0.` The angle between the sides is `(pi)/(3).` Find the perimeter of `Delta.`

Answer

To solve the problem step by step, we will follow the same logic as presented in the video transcript.

### Step 1: Find the roots of the quadratic equation
The given equation is:
\[ x^2 - 2\sqrt{3}x + 2 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -2\sqrt{3} \), and \( c = 2 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-2\sqrt{3})^2 - 4 \cdot 1 \cdot 2 = 12 - 8 = 4 \]

Now, substituting into the quadratic formula:
\[ x = \frac{2\sqrt{3} \pm \sqrt{4}}{2 \cdot 1} = \frac{2\sqrt{3} \pm 2}{2} = \sqrt{3} \pm 1 \]

Thus, the roots (sides of the triangle) are:
\[ A = \sqrt{3} + 1 \]
\[ B = \sqrt{3} - 1 \]

### Step 2: Calculate \( A + B \) and \( AB \)
Using the roots:
- \( A + B = (\sqrt{3} + 1) + (\sqrt{3} - 1) = 2\sqrt{3} \)
- \( AB = (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \)

### Step 3: Use the cosine rule to find the third side \( C \)
We know the angle \( C \) between sides \( A \) and \( B \) is \( \frac{\pi}{3} \) (or 60 degrees). The cosine of this angle is:
\[ \cos C = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Using the cosine rule:
\[ \cos C = \frac{A^2 + B^2 - C^2}{2AB} \]

Substituting the known values:
\[ \frac{1}{2} = \frac{A^2 + B^2 - C^2}{2 \cdot 2} \]
\[ 1 = A^2 + B^2 - C^2 \]
\[ C^2 = A^2 + B^2 - 1 \]

### Step 4: Calculate \( A^2 + B^2 \)
Using the identity:
\[ A^2 + B^2 = (A + B)^2 - 2AB \]
Substituting the values:
\[ A^2 + B^2 = (2\sqrt{3})^2 - 2 \cdot 2 = 12 - 4 = 8 \]

### Step 5: Substitute to find \( C^2 \)
Now substituting back:
\[ C^2 = 8 - 1 = 7 \]
Thus,
\[ C = \sqrt{7} \]

### Step 6: Calculate the perimeter of the triangle
The perimeter \( P \) of the triangle is given by:
\[ P = A + B + C = 2\sqrt{3} + \sqrt{7} \]

### Final Answer
The perimeter of the triangle is:
\[ P = 2\sqrt{3} + \sqrt{7} \]

---