If `tan^2 x + (1 - sqrt3) tan x - sqrt3 = 0` then `x=

Options

A. `n pi + pi/3`
B. `n pi - pi/3`
C. `n pi /pi/4`
D. `n pi-pi/4`

Answer

To solve the equation \( \tan^2 x + (1 - \sqrt{3}) \tan x - \sqrt{3} = 0 \), we can follow these steps:

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

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

### Step 3: Simplify the expression
Calculating \( (1 - \sqrt{3})^2 \):
\[ (1 - \sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3} \]
Now, calculate \( -4 \cdot 1 \cdot (-\sqrt{3}) = 4\sqrt{3} \).

Now, substituting back into the formula:
\[ \tan x = \frac{-(1 - \sqrt{3}) \pm \sqrt{(4 - 2\sqrt{3}) + 4\sqrt{3}}}{2} \]
\[ = \frac{-(1 - \sqrt{3}) \pm \sqrt{4 + 2\sqrt{3}}}{2} \]

### Step 4: Further simplify the square root
The expression under the square root simplifies to:
\[ 4 + 2\sqrt{3} = (2 + \sqrt{3})^2 \]
Thus, we have:
\[ \tan x = \frac{-(1 - \sqrt{3}) \pm (2 + \sqrt{3})}{2} \]

### Step 5: Calculate the two possible values for \( \tan x \)
1. For the positive case:
\[ \tan x = \frac{-(1 - \sqrt{3}) + (2 + \sqrt{3})}{2} = \frac{1 + 2\sqrt{3}}{2} \]
2. For the negative case:
\[ \tan x = \frac{-(1 - \sqrt{3}) - (2 + \sqrt{3})}{2} = \frac{-3}{2} \]

### Step 6: Find the angles corresponding to these tangent values
1. For \( \tan x = \frac{1 + 2\sqrt{3}}{2} \):
- This value corresponds to \( x = \tan^{-1}\left(\frac{1 + 2\sqrt{3}}{2}\right) \).

2. For \( \tan x = -\frac{3}{2} \):
- This value corresponds to \( x = \tan^{-1}\left(-\frac{3}{2}\right) \).

### Step 7: General solution
The general solutions for \( x \) can be expressed as:
1. \( x = n\pi + \tan^{-1}\left(\frac{1 + 2\sqrt{3}}{2}\right) \)
2. \( x = n\pi + \tan^{-1}\left(-\frac{3}{2}\right) \)

### Final Answer
Thus, the values of \( x \) are:
\[ x = n\pi + \tan^{-1}\left(\frac{1 + 2\sqrt{3}}{2}\right), \quad x = n\pi + \tan^{-1}\left(-\frac{3}{2}\right) \]