Solve: `sqrt3 x^(2)-2 sqrt2x- 2 sqrt3= 0`

Answer

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

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

### Step 2: Calculate the product \( ac \)
We need to calculate \( ac \):
\[ ac = \sqrt{3} \times (-2\sqrt{3}) = -6 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( -6 \) and add to \( -2\sqrt{2} \). The numbers are \( -3\sqrt{2} \) and \( \sqrt{2} \) because:
\[ -3\sqrt{2} \times \sqrt{2} = -6 \quad \text{and} \quad -3\sqrt{2} + \sqrt{2} = -2\sqrt{2} \]

### Step 4: Rewrite the middle term
We can rewrite the equation by splitting the middle term:
\[ \sqrt{3} x^2 - 3\sqrt{2} x + \sqrt{2} x - 2\sqrt{3} = 0 \]

### Step 5: Factor by grouping
Now, we will group the terms:
\[ (\sqrt{3} x^2 - 3\sqrt{2} x) + (\sqrt{2} x - 2\sqrt{3}) = 0 \]
Factoring out common terms:
\[ \sqrt{3} x (x - \sqrt{2}) + \sqrt{2} (x - \sqrt{3}) = 0 \]

### Step 6: Factor completely
We can factor this as:
\[ (\sqrt{3} x + \sqrt{2})(x - \sqrt{2}) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor to zero:
1. \( \sqrt{3} x + \sqrt{2} = 0 \)
2. \( x - \sqrt{2} = 0 \)

### Step 8: Solve for \( x \)
From the first factor:
\[ \sqrt{3} x = -\sqrt{2} \implies x = -\frac{\sqrt{2}}{\sqrt{3}} = -\frac{\sqrt{6}}{3} \]

From the second factor:
\[ x = \sqrt{2} \]

### Final Solutions
Thus, the solutions to the equation are:
\[ x = -\frac{\sqrt{6}}{3} \quad \text{and} \quad x = \sqrt{2} \]