Find zeroes of the given quadratic polynomials and verify the relation between zeroes and coefficients :
`4x^(2)-4x-3`

Explanation

To find the zeroes of the quadratic polynomial \(4x^2 - 4x - 3\) and verify the relationship between the zeroes and coefficients, we will follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the standard form \(ax^2 + bx + c\).
- Here, \(a = 4\), \(b = -4\), and \(c = -3\).

### Step 2: Use the quadratic formula
The zeroes of the quadratic polynomial can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \(D = b^2 - 4ac\):
\[ D = (-4)^2 - 4 \cdot 4 \cdot (-3) = 16 + 48 = 64 \]

### Step 4: Substitute values into the quadratic formula
Now, substituting the values of \(a\), \(b\), and \(D\) into the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 4} \]
\[ x = \frac{4 \pm 8}{8} \]

### Step 5: Calculate the zeroes
Now we calculate the two possible values for \(x\):
1. For the positive case:
\[ x_1 = \frac{4 + 8}{8} = \frac{12}{8} = \frac{3}{2} \]
2. For the negative case:
\[ x_2 = \frac{4 - 8}{8} = \frac{-4}{8} = -\frac{1}{2} \]

Thus, the zeroes of the polynomial \(4x^2 - 4x - 3\) are:
\[ x_1 = \frac{3}{2}, \quad x_2 = -\frac{1}{2} \]

### Step 6: Verify the relationship between zeroes and coefficients
The sum of the zeroes \((\alpha + \beta)\) and the product of the zeroes \((\alpha \beta)\) can be verified using the formulas:
- Sum of the zeroes: \(\alpha + \beta = -\frac{b}{a}\)
- Product of the zeroes: \(\alpha \beta = \frac{c}{a}\)

Calculating the sum:
\[ \alpha + \beta = \frac{3}{2} + \left(-\frac{1}{2}\right) = \frac{3 - 1}{2} = \frac{2}{2} = 1 \]
Calculating \(-\frac{b}{a}\):
\[ -\frac{b}{a} = -\frac{-4}{4} = 1 \]

Calculating the product:
\[ \alpha \beta = \frac{3}{2} \cdot \left(-\frac{1}{2}\right) = -\frac{3}{4} \]
Calculating \(\frac{c}{a}\):
\[ \frac{c}{a} = \frac{-3}{4} \]

### Conclusion
Both relationships are verified:
- Sum of the zeroes: \(\alpha + \beta = 1\) and \(-\frac{b}{a} = 1\)
- Product of the zeroes: \(\alpha \beta = -\frac{3}{4}\) and \(\frac{c}{a} = -\frac{3}{4}\)