The roots of the equation `3x^(2) - 4sqrt(3x) + 4 = 0` are

Answer

To find the roots of the quadratic equation \(3x^2 - 4\sqrt{3}x + 4 = 0\), 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 these values into the discriminant formula:
\[ D = 48 - 48 = 0 \]

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

### Step 4: Calculate the roots
The formula for the roots of a quadratic equation when \(D = 0\) is:
\[ x = \frac{-B}{2A} \]
Substituting the values of \(B\) and \(A\):
\[ x = \frac{-(-4\sqrt{3})}{2 \cdot 3} = \frac{4\sqrt{3}}{6} \]
Simplifying this:
\[ x = \frac{2\sqrt{3}}{3} \]

### Conclusion
The roots of the equation \(3x^2 - 4\sqrt{3}x + 4 = 0\) are:
\[ x = \frac{2\sqrt{3}}{3} \quad \text{(double root)} \]

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