If `3//2` and 4 are the two roots of a quadratic equation, then which one of the following is the correct quadratic equation?

Options

A. `2x^2-11xxx+12=0`
B. `6x^2-33xxx+18=0`
C. `-10xx^2-55xx-30=0`
D. `4x^2-41xx+24=0`

Answer

To find the quadratic equation given the roots \( \frac{3}{2} \) and \( 4 \), we can follow these steps:

### Step 1: Identify the roots
The roots of the quadratic equation are given as:
- \( \alpha = \frac{3}{2} \)
- \( \beta = 4 \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = \frac{3}{2} + 4 = \frac{3}{2} + \frac{8}{2} = \frac{11}{2} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) is calculated as:
\[ \alpha \beta = \frac{3}{2} \times 4 = 6 \]

### Step 4: Form the quadratic equation
Using the standard form of a quadratic equation with roots \( \alpha \) and \( \beta \):
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we calculated:
\[ x^2 - \left(\frac{11}{2}\right)x + 6 = 0 \]

### Step 5: Eliminate the fraction
To eliminate the fraction, we can multiply the entire equation by 2:
\[ 2x^2 - 11x + 12 = 0 \]

### Conclusion
The correct quadratic equation is:
\[ 2x^2 - 11x + 12 = 0 \]

### Final Answer
Thus, the correct option is **A: \( 2x^2 - 11x + 12 = 0 \)**.

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