If the zeroes of the quadratic equation are `(-1)/(2)` and 3/2, then the equation is

Options

A. `4x ^(2) - 6x + 3 =0`
B. `4x ^(2) - 5x + 3 =0`
C. `4x ^(2) - 7x + 3 =0`
D. `4x ^(2) - 4x - 3 =0`

Answer

To find the quadratic equation given its zeros, we can use the relationship between the zeros (roots) and the coefficients of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the Zeros**:
The zeros given are \( \alpha = -\frac{1}{2} \) and \( \beta = \frac{3}{2} \).

2. **Calculate the Sum of the Zeros**:
The sum of the zeros \( S \) is given by:
\[ S = \alpha + \beta = -\frac{1}{2} + \frac{3}{2} = \frac{2}{2} = 1 \]

3. **Calculate the Product of the Zeros**:
The product of the zeros \( P \) is given by:
\[ P = \alpha \cdot \beta = -\frac{1}{2} \cdot \frac{3}{2} = -\frac{3}{4} \]

4. **Form the Quadratic Equation**:
The standard form of a quadratic equation with roots \( \alpha \) and \( \beta \) is:
\[ x^2 - Sx + P = 0 \]
Substituting the values of \( S \) and \( P \):
\[ x^2 - 1x - \frac{3}{4} = 0 \]

5. **Clear the Fraction**:
To eliminate the fraction, multiply the entire equation by 4:
\[ 4(x^2 - x - \frac{3}{4}) = 0 \implies 4x^2 - 4x - 3 = 0 \]

6. **Final Quadratic Equation**:
Thus, the quadratic equation is:
\[ 4x^2 - 4x - 3 = 0 \]

### Final Answer:
The quadratic equation is \( 4x^2 - 4x - 3 = 0 \).

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