The least integral value of `'a'` for which the equation `x^2+2(a - 1)x + (2a + 1) = 0` has both the roots positive, is

Options

A. 3
B. 4
C. 1
D. 5

Explanation

To find the least integral value of \( a \) for which the equation

\[ x^2 + 2(a - 1)x + (2a + 1) = 0 \]

has both roots positive, we will follow these steps:

### Step 1: Identify the conditions for positive roots
For the quadratic equation \( ax^2 + bx + c = 0 \) to have both roots positive, we need to satisfy three conditions:
1. The discriminant \( D \) must be greater than or equal to 0.
2. The x-coordinate of the vertex must be positive.
3. The value of the function at \( x = 0 \) must be positive.

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

For our equation, \( a = 1 \), \( b = 2(a - 1) \), and \( c = 2a + 1 \).

Calculating \( D \):

\[ D = [2(a - 1)]^2 - 4 \cdot 1 \cdot (2a + 1) \]
\[ D = 4(a - 1)^2 - 4(2a + 1) \]
\[ D = 4[(a - 1)^2 - (2a + 1)] \]
\[ D = 4[a^2 - 2a + 1 - 2a - 1] \]
\[ D = 4[a^2 - 4a] \]
\[ D = 4a(a - 4) \]

For the roots to be real, we need \( D \geq 0 \):

\[ 4a(a - 4) \geq 0 \]

This gives us the intervals:

\[ a \leq 0 \quad \text{or} \quad a \geq 4 \]

### Step 3: Find the x-coordinate of the vertex
The x-coordinate of the vertex for a quadratic equation is given by:

\[ x = -\frac{b}{2a} \]

For our equation, this becomes:

\[ x = -\frac{2(a - 1)}{2 \cdot 1} = -(a - 1) = 1 - a \]

For the vertex to be positive:

\[ 1 - a > 0 \implies a < 1 \]

### Step 4: Evaluate \( f(0) \)
Next, we need to ensure that \( f(0) > 0 \):

\[ f(0) = 2a + 1 > 0 \]
\[ 2a > -1 \implies a > -\frac{1}{2} \]

### Step 5: Combine the conditions
Now we combine the conditions derived from the three steps:

1. From the discriminant: \( a \leq 0 \) or \( a \geq 4 \)
2. From the vertex: \( a < 1 \)
3. From \( f(0) \): \( a > -\frac{1}{2} \)

The only feasible solution that satisfies all conditions is:

\[ a \geq 4 \]

### Conclusion
The least integral value of \( a \) that satisfies all conditions is:

\[ \boxed{4} \]