Find zeroes of the given quadratic polynomials and verify the relation between zeroes and coefficients :
`x^(2)+2sqrt(2)x-6`

Explanation

To find the zeroes of the quadratic polynomial \( p(x) = x^2 + 2\sqrt{2}x - 6 \) and verify the relation between the zeroes and coefficients, we can follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic polynomial is given by:
\[ p(x) = ax^2 + bx + c \]
From the polynomial \( p(x) = x^2 + 2\sqrt{2}x - 6 \), we can identify:
- \( a = 1 \)
- \( b = 2\sqrt{2} \)
- \( c = -6 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (2\sqrt{2})^2 - 4 \cdot 1 \cdot (-6) \]
Calculating further:
\[ D = 8 + 24 = 32 \]

### Step 3: Find the zeroes using the quadratic formula
The zeroes (roots) of the quadratic polynomial are given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-2\sqrt{2} \pm \sqrt{32}}{2 \cdot 1} \]
Since \( \sqrt{32} = 4\sqrt{2} \), we can write:
\[ x = \frac{-2\sqrt{2} \pm 4\sqrt{2}}{2} \]
This simplifies to:
\[ x = \frac{(-2\sqrt{2} + 4\sqrt{2})}{2} \quad \text{and} \quad x = \frac{(-2\sqrt{2} - 4\sqrt{2})}{2} \]
Calculating these gives:
\[ x_1 = \frac{2\sqrt{2}}{2} = \sqrt{2} \quad \text{and} \quad x_2 = \frac{-6\sqrt{2}}{2} = -3\sqrt{2} \]

### Step 4: Verify the relation between zeroes and coefficients
The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2\sqrt{2}}{1} = -2\sqrt{2} \]
Calculating the sum of the roots we found:
\[ \sqrt{2} + (-3\sqrt{2}) = -2\sqrt{2} \]
This confirms the sum of the roots.

The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-6}{1} = -6 \]
Calculating the product of the roots we found:
\[ \sqrt{2} \cdot (-3\sqrt{2}) = -3 \cdot 2 = -6 \]
This confirms the product of the roots.

### Conclusion
Thus, we have found the zeroes of the polynomial \( p(x) = x^2 + 2\sqrt{2}x - 6 \) to be \( \sqrt{2} \) and \( -3\sqrt{2} \), and verified the relations:
- Sum of the roots: \( -2\sqrt{2} \)
- Product of the roots: \( -6 \)