The equation whose roots are half of roots of the equation `2x^(2) + 5x + 4= 0` is

Options

A. `8x^(2)+10x+1=0`
B. `4x^(2)+5x+2=0`
C. `8x^(2)+5x+1=0`
D. `16x^(2)+5x+1=0`

Answer

To find the equation whose roots are half of the roots of the given quadratic equation \(2x^2 + 5x + 4 = 0\), we can follow these steps:

### Step 1: Identify the roots of the given equation
The given quadratic equation is:
\[ 2x^2 + 5x + 4 = 0 \]
Let the roots of this equation be \(\alpha\) and \(\beta\).

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \((\alpha + \beta)\) is given by:
\[ \alpha + \beta = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{5}{2} \]

- The product of the roots \((\alpha \beta)\) is given by:
\[ \alpha \beta = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{4}{2} = 2 \]

### Step 3: Determine the new roots
We need to find the equation whose roots are half of the original roots. Therefore, the new roots \(\alpha'\) and \(\beta'\) are:
\[ \alpha' = \frac{\alpha}{2}, \quad \beta' = \frac{\beta}{2} \]

### Step 4: Calculate the sum and product of the new roots
- The sum of the new roots \((\alpha' + \beta')\) is:
\[ \alpha' + \beta' = \frac{\alpha}{2} + \frac{\beta}{2} = \frac{\alpha + \beta}{2} = \frac{-\frac{5}{2}}{2} = -\frac{5}{4} \]

- The product of the new roots \((\alpha' \beta')\) is:
\[ \alpha' \beta' = \frac{\alpha}{2} \cdot \frac{\beta}{2} = \frac{\alpha \beta}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Form the new quadratic equation
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - (\alpha' + \beta')x + \alpha' \beta' = 0 \]
Substituting the values we found:
\[ x^2 - \left(-\frac{5}{4}\right)x + \frac{1}{2} = 0 \]
This simplifies to:
\[ x^2 + \frac{5}{4}x + \frac{1}{2} = 0 \]

### Step 6: Eliminate fractions by multiplying through by 4
To eliminate the fractions, we multiply the entire equation by 4:
\[ 4x^2 + 5x + 2 = 0 \]

### Conclusion
The required quadratic equation whose roots are half of the roots of the original equation is:
\[ \boxed{4x^2 + 5x + 2 = 0} \]