Solve the following equation: `tan^2x+(1-sqrt(3))tanx-sqrt(3)=0`

Explanation

To solve the equation \( \tan^2 x + (1 - \sqrt{3}) \tan x - \sqrt{3} = 0 \), we can treat it as a quadratic equation in terms of \( \tan x \). Let's denote \( y = \tan x \). The equation then becomes:

\[ y^2 + (1 - \sqrt{3})y - \sqrt{3} = 0 \]

### Step 1: Identify coefficients
In the standard form of a quadratic equation \( ay^2 + by + c = 0 \), we have:
- \( a = 1 \)
- \( b = 1 - \sqrt{3} \)
- \( c = -\sqrt{3} \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (1 - \sqrt{3})^2 - 4 \cdot 1 \cdot (-\sqrt{3}) \]

Calculating \( (1 - \sqrt{3})^2 \):

\[ (1 - \sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3} \]

Now substituting back into the discriminant:

\[ D = (4 - 2\sqrt{3}) + 4\sqrt{3} = 4 + 2\sqrt{3} \]

### Step 3: Use the quadratic formula
The roots of the quadratic equation are given by:

\[ y = \frac{-b \pm \sqrt{D}}{2a} \]

Substituting the values of \( b \), \( D \), and \( a \):

\[ y = \frac{-(1 - \sqrt{3}) \pm \sqrt{4 + 2\sqrt{3}}}{2} \]

This simplifies to:

\[ y = \frac{-1 + \sqrt{3} \pm \sqrt{4 + 2\sqrt{3}}}{2} \]

### Step 4: Simplify the expression
Now we need to simplify \( \sqrt{4 + 2\sqrt{3}} \). We can rewrite it as:

\[ \sqrt{4 + 2\sqrt{3}} = \sqrt{(\sqrt{3} + 1)^2} = \sqrt{3} + 1 \]

Now substituting back into the expression for \( y \):

\[ y = \frac{-1 + \sqrt{3} \pm (\sqrt{3} + 1)}{2} \]

This gives us two cases:

1. \( y = \frac{-1 + \sqrt{3} + \sqrt{3} + 1}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \)
2. \( y = \frac{-1 + \sqrt{3} - \sqrt{3} - 1}{2} = \frac{-2}{2} = -1 \)

### Step 5: Solve for \( x \)
Now we have two values for \( y \):

1. \( \tan x = \sqrt{3} \)
2. \( \tan x = -1 \)

For \( \tan x = \sqrt{3} \):
- The general solution is \( x = n\pi + \frac{\pi}{3} \) where \( n \) is any integer.

For \( \tan x = -1 \):
- The general solution is \( x = n\pi - \frac{\pi}{4} \) where \( n \) is any integer.

### Final Solutions
Thus, the complete solution set is:

\[ x = n\pi + \frac{\pi}{3} \quad \text{and} \quad x = n\pi - \frac{\pi}{4} \]