Solve `x^(2)-(sqrt3+1)x+sqrt3=0`.

Answer

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

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

### Step 2: Use 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 \( c \) into the formula:

### Step 3: Calculate \( b^2 - 4ac \)
First, we need to calculate \( b^2 - 4ac \):
\[ b^2 = (-(\sqrt{3} + 1))^2 = (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
Next, calculate \( 4ac \):
\[ 4ac = 4 \cdot 1 \cdot \sqrt{3} = 4\sqrt{3} \]
Now, substitute these into the discriminant:
\[ b^2 - 4ac = (4 + 2\sqrt{3}) - 4\sqrt{3} = 4 - 2\sqrt{3} \]

### Step 4: Substitute back into the quadratic formula
Now, substituting \( b \) and the discriminant back into the quadratic formula:
\[ x = \frac{-(-(\sqrt{3} + 1)) \pm \sqrt{4 - 2\sqrt{3}}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{\sqrt{3} + 1 \pm \sqrt{4 - 2\sqrt{3}}}{2} \]

### Step 5: Simplify \( \sqrt{4 - 2\sqrt{3}} \)
To simplify \( \sqrt{4 - 2\sqrt{3}} \), we can express it in a different form. Let’s assume it can be expressed as \( \sqrt{a} - \sqrt{b} \). Squaring both sides gives:
\[ 4 - 2\sqrt{3} = a + b - 2\sqrt{ab} \]
From this, we can equate:
1. \( a + b = 4 \)
2. \( -2\sqrt{ab} = -2\sqrt{3} \) ⇒ \( \sqrt{ab} = \sqrt{3} \) ⇒ \( ab = 3 \)

Solving these two equations:
Let \( a = 4 - b \). Then substituting into \( ab = 3 \):
\[ (4 - b)b = 3 \implies 4b - b^2 = 3 \implies b^2 - 4b + 3 = 0 \]
Factoring gives:
\[ (b - 3)(b - 1) = 0 \implies b = 3 \text{ or } b = 1 \]
Thus, \( a = 1 \) or \( a = 3 \). Therefore, we can express:
\[ \sqrt{4 - 2\sqrt{3}} = \sqrt{3} - 1 \]

### Step 6: Final substitution
Now substituting back:
\[ x = \frac{\sqrt{3} + 1 \pm (\sqrt{3} - 1)}{2} \]
This gives us two cases:
1. \( x = \frac{\sqrt{3} + 1 + \sqrt{3} - 1}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \)
2. \( x = \frac{\sqrt{3} + 1 - (\sqrt{3} - 1)}{2} = \frac{2}{2} = 1 \)

### Conclusion
Thus, the solutions to the equation \( x^2 - (\sqrt{3} + 1)x + \sqrt{3} = 0 \) are:
\[ x = \sqrt{3} \quad \text{and} \quad x = 1 \]