If `{2(1)/(4)}` is the solution set of a quadratic equation, find the equation.

Options

A. `4x^(2)-9x+2=0`
B. `2x^(2)-9x + 4=0`
C. `x^(2)-18x-6=0`
D. `2x^(2)-18x + 3= 0`

Answer

To find the quadratic equation given the solution set \( \{2, \frac{1}{4}\} \), we can follow these steps:

### Step 1: Identify the roots
Let the roots of the quadratic equation be \( \alpha = 2 \) and \( \beta = \frac{1}{4} \).

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be calculated as follows:
\[ \alpha + \beta = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated as:
\[ \alpha \beta = 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 4: Write the general form of the quadratic equation
The general form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

### Step 5: Substitute the values of the sum and product of the roots
Substituting the values we calculated:
\[ x^2 - \left(\frac{9}{4}\right)x + \frac{1}{2} = 0 \]

### Step 6: Eliminate fractions by multiplying through by 4
To eliminate the fractions, we multiply the entire equation by 4:
\[ 4x^2 - 9x + 2 = 0 \]

### Conclusion
Thus, the quadratic equation is:
\[ 4x^2 - 9x + 2 = 0 \]