Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
`2sqrt(3)x^(2)-5x+sqrt(3)=0`

Answer

To find the roots of the quadratic equation \(2\sqrt{3}x^2 - 5x + \sqrt{3} = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
We compare the given equation with the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
- Here, \(a = 2\sqrt{3}\), \(b = -5\), and \(c = \sqrt{3}\).

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-5)^2 - 4 \cdot (2\sqrt{3}) \cdot (\sqrt{3}) \]
Calculating it step-by-step:
\[ D = 25 - 4 \cdot 2\sqrt{3} \cdot \sqrt{3} \]
\[ D = 25 - 4 \cdot 2 \cdot 3 \]
\[ D = 25 - 24 \]
\[ D = 1 \]

### Step 3: Check the nature of the roots
Since \(D > 0\), the equation has two distinct real roots.

### Step 4: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \(b\), \(D\), and \(a\):
\[ x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot (2\sqrt{3})} \]
This simplifies to:
\[ x = \frac{5 \pm 1}{4\sqrt{3}} \]

### Step 5: Calculate the two roots
1. **First root (using the positive sign)**:
\[ x_1 = \frac{5 + 1}{4\sqrt{3}} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} \]
Rationalizing the denominator:
\[ x_1 = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2} \]

2. **Second root (using the negative sign)**:
\[ x_2 = \frac{5 - 1}{4\sqrt{3}} = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Conclusion
The roots of the equation \(2\sqrt{3}x^2 - 5x + \sqrt{3} = 0\) are:
\[ x_1 = \frac{\sqrt{3}}{2}, \quad x_2 = \frac{1}{\sqrt{3}} \]