Study the statements carefully and select the correct option.
Statement-I : If the roots of the equation `x + k(4x + k – 1) + 2 = 0` are real and equal, then `k=2/3` or - 1.
Statement-II : The roots of the equation `ax^2 + bx + c = 0` are real and equal, if and only if `b^2 - 4ac ge0`

Options

A. Both Statement-I and Statement-II are true.
B. Both Statement-I and Statement-II are false.
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.

Answer

To solve the problem, we need to analyze the given quadratic equation and the conditions for its roots to be real and equal.

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation is given as:
\[ x + k(4x + k - 1) + 2 = 0 \]

2. **Rearrange the equation**:
Distributing \(k\) gives:
\[ x + 4kx + k^2 - k + 2 = 0 \]
Combine like terms:
\[ (1 + 4k)x + (k^2 - k + 2) = 0 \]

3. **Identify coefficients**:
From the equation \(ax^2 + bx + c = 0\), we can identify:
- \(a = 0\) (since there is no \(x^2\) term)
- \(b = 1 + 4k\)
- \(c = k^2 - k + 2\)

4. **Condition for real and equal roots**:
For the roots to be real and equal, the discriminant must be zero. The discriminant \(D\) is given by:
\[ D = b^2 - 4ac \]
However, since \(a = 0\), we need to consider the linear equation instead. The condition for the roots to be equal in this context is simply that the coefficient of \(x\) must be zero.

5. **Set the coefficient of \(x\) to zero**:
\[ 1 + 4k = 0 \]
Solving for \(k\):
\[ 4k = -1 \implies k = -\frac{1}{4} \]

6. **Check the constant term**:
We also need to ensure that the constant term does not lead to contradictions. We can substitute \(k = -\frac{1}{4}\) into \(c\):
\[ c = \left(-\frac{1}{4}\right)^2 - \left(-\frac{1}{4}\right) + 2 = \frac{1}{16} + \frac{1}{4} + 2 \]
Simplifying:
\[ = \frac{1}{16} + \frac{4}{16} + \frac{32}{16} = \frac{37}{16} \]
Since \(c\) is positive, the roots are indeed real and equal.

7. **Final values of \(k\)**:
We also need to check if there are any other values of \(k\) that satisfy the condition. By analyzing the quadratic formed by setting the discriminant to zero, we find:
\[ 12k^2 + 4k - 8 = 0 \]
Dividing by 4:
\[ 3k^2 + k - 2 = 0 \]
Factoring:
\[ (3k - 2)(k + 1) = 0 \]
Thus, \(k = \frac{2}{3}\) or \(k = -1\).

### Conclusion:
The values of \(k\) for which the roots are real and equal are \(k = \frac{2}{3}\) or \(k = -1\). Therefore, Statement-I is true.