Solve each of the following quadratic equations:
`sqrt(3)x^(2)+11x+6sqrt(3)=0`

Explanation

To solve the quadratic equation \( \sqrt{3}x^2 + 11x + 6\sqrt{3} = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
Here,
- \( a = \sqrt{3} \)
- \( b = 11 \)
- \( c = 6\sqrt{3} \)

### Step 2: Calculate the product \( ac \)
We need to find the product \( ac \):
\[ ac = \sqrt{3} \times 6\sqrt{3} = 6 \times 3 = 18 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 18 \) and add up to \( 11 \). The numbers are \( 9 \) and \( 2 \) because:
\[ 9 \times 2 = 18 \quad \text{and} \quad 9 + 2 = 11 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the equation by splitting the middle term \( 11x \) into \( 9x + 2x \):
\[ \sqrt{3}x^2 + 9x + 2x + 6\sqrt{3} = 0 \]

### Step 5: Factor by grouping
Group the terms:
\[ (\sqrt{3}x^2 + 9x) + (2x + 6\sqrt{3}) = 0 \]
Now, factor out the common terms from each group:
\[ x(\sqrt{3}x + 9) + 2(\sqrt{3}x + 3) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out \( \sqrt{3}x + 9 \):
\[ (\sqrt{3}x + 9)(x + 2) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor to zero:
1. \( \sqrt{3}x + 9 = 0 \)
2. \( x + 2 = 0 \)

### Step 8: Solve for \( x \)
For the first equation:
\[ \sqrt{3}x + 9 = 0 \implies \sqrt{3}x = -9 \implies x = -\frac{9}{\sqrt{3}} = -3\sqrt{3} \]

For the second equation:
\[ x + 2 = 0 \implies x = -2 \]

### Final Solution
The roots of the quadratic equation are:
\[ x = -3\sqrt{3} \quad \text{and} \quad x = -2 \]

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