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

Options

A. real and unequal
B. rational and equal
C. irrational and equal
D. irrational and unequal

Answer

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

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 1\)
- \(b = -2\sqrt{3}\)
- \(c = 3\)

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

### Step 3: Simplify the discriminant
Calculating \(D\):
\[ D = (2\sqrt{3})^2 - 12 \]
\[ D = 4 \cdot 3 - 12 \]
\[ D = 12 - 12 \]
\[ D = 0 \]

### Step 4: Determine the nature of the roots
Since the discriminant \(D = 0\), this indicates that the quadratic equation has real and equal roots.

### Step 5: Calculate the roots using the quadratic formula
The roots of the quadratic equation can be found using the formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-2\sqrt{3}) \pm \sqrt{0}}{2 \cdot 1} \]
\[ x = \frac{2\sqrt{3}}{2} \]
\[ x = \sqrt{3} \]

### Conclusion
The roots of the equation \(x^2 - 2\sqrt{3}x + 3 = 0\) are both equal to \(\sqrt{3}\).

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