If the roots 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 roots, we can use the relationship between the roots and the coefficients of the equation. The general form of a quadratic equation is:

\[ ax^2 + bx + c = 0 \]

Where:
- \( a \) is the coefficient of \( x^2 \)
- \( b \) is the coefficient of \( x \)
- \( c \) is the constant term

### Step-by-Step Solution:

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

2. **Calculate the Sum of the Roots:**
The sum of the roots \( S \) is calculated as:
\[ S = \alpha + \beta = -\frac{1}{2} + \frac{3}{2} \]
Simplifying this:
\[ S = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

3. **Calculate the Product of the Roots:**
The product of the roots \( P \) is calculated as:
\[ P = \alpha \cdot \beta = -\frac{1}{2} \cdot \frac{3}{2} \]
Simplifying this:
\[ P = -\frac{3}{4} \]

4. **Form the Quadratic Equation:**
Using the standard form of the quadratic equation:
\[ x^2 - (S)x + P = 0 \]
Substituting the values of \( S \) and \( P \):
\[ x^2 - (1)x - \frac{3}{4} = 0 \]

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

### Final Quadratic Equation:
The quadratic equation is:
\[ 4x^2 - 4x - 3 = 0 \]