Find zeroes of the quadratic `sqrt(3)x^(2)-8x+4sqrt(3)` and verify the relation between the zeroes and coefficients.

Explanation

To find the zeroes of the quadratic equation \( \sqrt{3}x^2 - 8x + 4\sqrt{3} \) and verify the relationship between the zeroes and coefficients, we can follow these steps:

### Step 1: Write the quadratic equation
Let \( p(x) = \sqrt{3}x^2 - 8x + 4\sqrt{3} \).

### Step 2: Factor the quadratic expression
To factor the quadratic, we look for two numbers that multiply to \( a \cdot c \) (where \( a = \sqrt{3} \) and \( c = 4\sqrt{3} \)) and add to \( b \) (where \( b = -8 \)).

1. Calculate \( a \cdot c = \sqrt{3} \cdot 4\sqrt{3} = 4 \cdot 3 = 12 \).
2. We need two numbers that multiply to \( 12 \) and add to \( -8 \). The numbers are \( -6 \) and \( -2 \).

Thus, we can rewrite the quadratic as:
\[ p(x) = \sqrt{3}x^2 - 6x - 2x + 4\sqrt{3} \]

### Step 3: Group the terms
Now, we group the terms:
\[ = (\sqrt{3}x^2 - 6x) + (-2x + 4\sqrt{3}) \]

### Step 4: Factor by grouping
Now, we factor out common terms from each group:
\[ = x(\sqrt{3}x - 6) - 2(\sqrt{3}x - 2) \]

Now, we can factor out \( (\sqrt{3}x - 2) \):
\[ = (\sqrt{3}x - 2)(x - 2\sqrt{3}) \]

### Step 5: Set the factors to zero
To find the zeroes, we set each factor to zero:
1. \( \sqrt{3}x - 2 = 0 \)
2. \( x - 2\sqrt{3} = 0 \)

Solving these gives:
1. \( \sqrt{3}x = 2 \) → \( x = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)
2. \( x = 2\sqrt{3} \)

### Step 6: List the zeroes
The zeroes of the quadratic are:
1. \( x_1 = 2\sqrt{3} \)
2. \( x_2 = \frac{2\sqrt{3}}{3} \)

### Step 7: Verify the relationship between the zeroes and coefficients
For a quadratic \( ax^2 + bx + c \), the relationships are:
- Sum of the zeroes \( x_1 + x_2 = -\frac{b}{a} \)
- Product of the zeroes \( x_1 \cdot x_2 = \frac{c}{a} \)

#### Calculate the sum of the zeroes:
\[ x_1 + x_2 = 2\sqrt{3} + \frac{2\sqrt{3}}{3} \]
Finding a common denominator:
\[ = \frac{6\sqrt{3}}{3} + \frac{2\sqrt{3}}{3} = \frac{8\sqrt{3}}{3} \]

Now, calculate \( -\frac{b}{a} \):
\[ -\frac{-8}{\sqrt{3}} = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \]
Thus, the sum of the zeroes is verified.

#### Calculate the product of the zeroes:
\[ x_1 \cdot x_2 = 2\sqrt{3} \cdot \frac{2\sqrt{3}}{3} = \frac{4 \cdot 3}{3} = 4 \]

Now, calculate \( \frac{c}{a} \):
\[ \frac{4\sqrt{3}}{\sqrt{3}} = 4 \]
Thus, the product of the zeroes is verified.

### Final Result
The zeroes of the quadratic equation \( \sqrt{3}x^2 - 8x + 4\sqrt{3} \) are \( 2\sqrt{3} \) and \( \frac{2\sqrt{3}}{3} \), and the relationships between the zeroes and coefficients are verified.

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