The values of a for which the equation `2x^(2) -2(2a+1) x+a(a+1) = 0` may have one root less them a and other root greater than a are given by

Options

A. `1 gt a gt 0`
B. `-1 lt a lt 0`
C. `a ge 0`
D. `a gt 0 or a lt -1`

Explanation

To solve the problem step by step, we need to analyze the given quadratic equation and determine the conditions under which it has one root less than \( a \) and the other root greater than \( a \).

### Step 1: Identify the quadratic equation
The quadratic equation given is:
\[ 2x^2 - 2(2a + 1)x + a(a + 1) = 0 \]

### Step 2: Define the coefficients
From the equation, we can identify the coefficients:
- \( A = 2 \)
- \( B = -2(2a + 1) \)
- \( C = a(a + 1) \)

### Step 3: Find the discriminant
To determine the nature of the roots, we calculate the discriminant \( D \):
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = [-2(2a + 1)]^2 - 4 \cdot 2 \cdot a(a + 1) \]
\[ D = 4(2a + 1)^2 - 8a(a + 1) \]
Expanding this:
\[ D = 4(4a^2 + 4a + 1) - 8(a^2 + a) \]
\[ D = 16a^2 + 16a + 4 - 8a^2 - 8a \]
\[ D = 8a^2 + 8a + 4 \]

### Step 4: Set the discriminant greater than zero
For the quadratic equation to have two distinct roots, we need:
\[ D > 0 \]
Thus, we need:
\[ 8a^2 + 8a + 4 > 0 \]
Dividing the entire inequality by 4:
\[ 2a^2 + 2a + 1 > 0 \]

### Step 5: Analyze the quadratic inequality
The quadratic \( 2a^2 + 2a + 1 \) has a discriminant:
\[ D' = 2^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 \]
Since the discriminant is negative, the quadratic \( 2a^2 + 2a + 1 \) is always greater than zero for all \( a \in \mathbb{R} \).

### Step 6: Evaluate the function at \( a \)
Next, we need to check the second condition: \( f(a) < 0 \).
\[ f(a) = 2a^2 - 2(2a + 1)a + a(a + 1) \]
Simplifying:
\[ f(a) = 2a^2 - (4a^2 + 2a) + (a^2 + a) \]
\[ f(a) = 2a^2 - 4a^2 - 2a + a^2 + a \]
\[ f(a) = -a^2 - a < 0 \]
This implies:
\[ a^2 + a > 0 \]

### Step 7: Factor the inequality
Factoring gives:
\[ a(a + 1) > 0 \]
The solutions to this inequality are:
1. \( a > 0 \)
2. \( a < -1 \)

### Step 8: Combine the results
Thus, the values of \( a \) for which the quadratic equation has one root less than \( a \) and the other root greater than \( a \) are:
\[ a > 0 \quad \text{or} \quad a < -1 \]

### Final Answer
The correct option is:
**Option D: \( a > 0 \) or \( a < -1 \)**