If `alpha, beta` are the roots of `2x^(2) + 5x + 2 = 0`, then the equation with roots `(1)/(alpha+1), (1)/(beta+1)` is

Options

A. `x^(2) + x - 2 = 0`
B. `x^(2) - x - 2 = 0`
C. `x^(2) + x - 1 = 0`
D. `x^(2) + x - 3 = 0`

Answer

To find the equation whose roots are \( \frac{1}{\alpha + 1} \) and \( \frac{1}{\beta + 1} \), we will follow these steps:

### Step 1: Identify the coefficients of the given quadratic equation
The given equation is:
\[ 2x^2 + 5x + 2 = 0 \]

Here, we can identify:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 2 \)

### Step 2: Calculate the sum and product of the roots \( \alpha \) and \( \beta \)
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{5}{2} \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{2}{2} = 1 \]

### Step 3: Find the new roots \( \frac{1}{\alpha + 1} \) and \( \frac{1}{\beta + 1} \)
To find the new roots, we need to express them in terms of \( \alpha \) and \( \beta \):
\[ \frac{1}{\alpha + 1} \quad \text{and} \quad \frac{1}{\beta + 1} \]

### Step 4: Calculate the sum of the new roots
The sum of the new roots is:
\[ \frac{1}{\alpha + 1} + \frac{1}{\beta + 1} = \frac{(\beta + 1) + (\alpha + 1)}{(\alpha + 1)(\beta + 1)} = \frac{\alpha + \beta + 2}{(\alpha + 1)(\beta + 1)} \]
Substituting the values we calculated:
\[ \alpha + \beta + 2 = -\frac{5}{2} + 2 = -\frac{5}{2} + \frac{4}{2} = -\frac{1}{2} \]
Now, we need to calculate \( (\alpha + 1)(\beta + 1) \):
\[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 = 1 - \frac{5}{2} + 1 = 1 - \frac{5}{2} + \frac{2}{2} = 1 - \frac{3}{2} = -\frac{1}{2} \]
Thus, the sum of the new roots becomes:
\[ \frac{-\frac{1}{2}}{-\frac{1}{2}} = 1 \]

### Step 5: Calculate the product of the new roots
The product of the new roots is:
\[ \frac{1}{(\alpha + 1)(\beta + 1)} = \frac{1}{-\frac{1}{2}} = -2 \]

### Step 6: Form the new quadratic equation
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values:
\[ x^2 - (1)x + (-2) = 0 \quad \Rightarrow \quad x^2 - x - 2 = 0 \]

### Final Answer
The equation with roots \( \frac{1}{\alpha + 1} \) and \( \frac{1}{\beta + 1} \) is:
\[ \boxed{x^2 - x - 2 = 0} \]