Find the nature of the roots of the following quadratic equations. If the real roots exist, find them :
`3x^(2) - 4 sqrt(3)x + 4 = 0`

Answer

To solve the quadratic equation \(3x^2 - 4\sqrt{3}x + 4 = 0\) and find the nature of its roots, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 3\)
- \(b = -4\sqrt{3}\)
- \(c = 4\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-4\sqrt{3})^2 - 4 \cdot 3 \cdot 4 \]
Calculating \(b^2\):
\[ (-4\sqrt{3})^2 = 16 \cdot 3 = 48 \]
Calculating \(4ac\):
\[ 4 \cdot 3 \cdot 4 = 48 \]
Now substituting back into the discriminant formula:
\[ D = 48 - 48 = 0 \]

### Step 3: Determine the nature of the roots
Since the discriminant \(D = 0\), this indicates that the quadratic equation has real and equal roots.

### Step 4: Find the roots using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Since \(D = 0\), the formula simplifies to:
\[ x = \frac{-b}{2a} \]
Substituting \(b = -4\sqrt{3}\) and \(a = 3\):
\[ x = \frac{-(-4\sqrt{3})}{2 \cdot 3} = \frac{4\sqrt{3}}{6} = \frac{2\sqrt{3}}{3} \]

### Step 5: State the roots
Since the roots are equal, we have:
\[ x_1 = x_2 = \frac{2\sqrt{3}}{3} \]

### Summary
The nature of the roots of the quadratic equation \(3x^2 - 4\sqrt{3}x + 4 = 0\) is that they are real and equal. The roots are:
\[ x = \frac{2\sqrt{3}}{3} \]