If ` alpha , beta ` are the roots of ` x^(2) - 2 x + 4 = 0 " then " (alpha )/(beta)` is equal to

Options

A. `(1)/(2) (1 - sqrt(3i))`
B. `-(1)/(2)(1+ sqrt(3))`
C. `(1)/(2)(- 1 pm sqrt(3i))`
D. `(1)/(2) ( sqrt(3)-i)`

Answer

To find the value of \(\frac{\alpha}{\beta}\) where \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 2x + 4 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is:
\[ x^2 - 2x + 4 = 0 \]
Here, \(a = 1\), \(b = -2\), and \(c = 4\).

### Step 2: Use the quadratic formula to find the roots
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{2 \pm \sqrt{4 - 16}}{2} \]
\[ x = \frac{2 \pm \sqrt{-12}}{2} \]
\[ x = \frac{2 \pm 2i\sqrt{3}}{2} \]
\[ x = 1 \pm i\sqrt{3} \]

### Step 3: Identify the roots
Thus, the roots are:
\[ \alpha = 1 + i\sqrt{3}, \quad \beta = 1 - i\sqrt{3} \]

### Step 4: Calculate \(\frac{\alpha}{\beta}\)
Now we calculate:
\[ \frac{\alpha}{\beta} = \frac{1 + i\sqrt{3}}{1 - i\sqrt{3}} \]

### Step 5: Multiply by the conjugate
To simplify this expression, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{\alpha}{\beta} = \frac{(1 + i\sqrt{3})(1 + i\sqrt{3})}{(1 - i\sqrt{3})(1 + i\sqrt{3})} \]

### Step 6: Simplify the denominator
The denominator simplifies as follows:
\[ (1 - i\sqrt{3})(1 + i\sqrt{3}) = 1^2 - (i\sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

### Step 7: Simplify the numerator
Now simplify the numerator:
\[ (1 + i\sqrt{3})^2 = 1^2 + 2 \cdot 1 \cdot i\sqrt{3} + (i\sqrt{3})^2 = 1 + 2i\sqrt{3} - 3 = -2 + 2i\sqrt{3} \]

### Step 8: Combine the results
Now we have:
\[ \frac{\alpha}{\beta} = \frac{-2 + 2i\sqrt{3}}{4} = \frac{-1 + i\sqrt{3}}{2} \]

### Final Result
Thus, the value of \(\frac{\alpha}{\beta}\) is:
\[ \frac{\alpha}{\beta} = \frac{-1 + i\sqrt{3}}{2} \]