From question number 1 to 16, find zeroes of the given quadratic polynomials and verify the relation between zeroes and coefficients :
`2x^(2)-15-11x`

Explanation

To find the zeroes of the quadratic polynomial \(2x^2 - 11x - 15\) and verify the relation between the zeroes and coefficients, we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \(2x^2 - 11x - 15\). Here, we identify the coefficients as follows:
- \(a = 2\)
- \(b = -11\)
- \(c = -15\)

### Step 2: Use the quadratic formula
The zeroes (roots) of a quadratic polynomial \(ax^2 + bx + c\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4 \cdot 2 \cdot (-15)}}{2 \cdot 2} \]

### Step 3: Calculate the discriminant
First, we calculate the discriminant \(D = b^2 - 4ac\):
\[ D = (-11)^2 - 4 \cdot 2 \cdot (-15) = 121 + 120 = 241 \]

### Step 4: Substitute the discriminant back into the formula
Now, substituting \(D\) back into the quadratic formula:
\[ x = \frac{11 \pm \sqrt{241}}{4} \]

### Step 5: Find the zeroes
Thus, the two zeroes (roots) are:
\[ x_1 = \frac{11 + \sqrt{241}}{4}, \quad x_2 = \frac{11 - \sqrt{241}}{4} \]

### Step 6: Verify the relation between zeroes and coefficients
The sum of the roots \(x_1 + x_2\) should equal \(-\frac{b}{a}\) and the product of the roots \(x_1 \cdot x_2\) should equal \(\frac{c}{a}\).

1. **Sum of the roots**:
\[ x_1 + x_2 = \frac{11 + \sqrt{241}}{4} + \frac{11 - \sqrt{241}}{4} = \frac{22}{4} = \frac{11}{2} \]
Now, check \(-\frac{b}{a}\):
\[ -\frac{-11}{2} = \frac{11}{2} \]

2. **Product of the roots**:
\[ x_1 \cdot x_2 = \left(\frac{11 + \sqrt{241}}{4}\right) \left(\frac{11 - \sqrt{241}}{4}\right) = \frac{121 - 241}{16} = \frac{-120}{16} = -\frac{15}{2} \]
Now, check \(\frac{c}{a}\):
\[ \frac{-15}{2} = -\frac{15}{2} \]

Both relations hold true.

### Final Answer:
The zeroes of the polynomial \(2x^2 - 11x - 15\) are:
\[ x_1 = \frac{11 + \sqrt{241}}{4}, \quad x_2 = \frac{11 - \sqrt{241}}{4} \]
And the relations between the zeroes and coefficients are verified.