If both the distinct roots of the equation `|sinx|^2+|sinx|+b=0in[0,pi]` are real, then the values of `b` are `[-2,0]` (b) `(-2,0)` `[-2,0]` (d) `non eoft h e s e`

Options

A. `[-2, 0]`
B. `(-2, 0)`
C. `[-2, 0)`
D. none of these

Explanation

To solve the equation \(|\sin x|^2 + |\sin x| + b = 0\) for \(x\) in the interval \([0, \pi]\) and to find the values of \(b\) such that both distinct roots are real, we can follow these steps:

### Step 1: Rewrite the equation
The equation can be rewritten as:
\[ |\sin x|^2 + |\sin x| + b = 0 \]
Let \(y = |\sin x|\). The equation then becomes:
\[ y^2 + y + b = 0 \]

### Step 2: Apply the quadratic formula
To find the roots of the quadratic equation \(y^2 + y + b = 0\), we use the quadratic formula:
\[ y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
where \(A = 1\), \(B = 1\), and \(C = b\). Thus, we have:
\[ y = \frac{-1 \pm \sqrt{1 - 4b}}{2} \]

### Step 3: Ensure the roots are real and distinct
For the roots to be real and distinct, the discriminant must be positive:
\[ 1 - 4b > 0 \]
This simplifies to:
\[ 1 > 4b \quad \Rightarrow \quad b < \frac{1}{4} \]

### Step 4: Consider the range of \(y\)
Since \(y = |\sin x|\) and \(x\) is in the interval \([0, \pi]\), \(y\) can take values in the range \([0, 1]\). Therefore, we need both roots to fall within this range.

### Step 5: Set conditions for the roots
The roots are given by:
\[ y_1 = \frac{-1 + \sqrt{1 - 4b}}{2} \quad \text{and} \quad y_2 = \frac{-1 - \sqrt{1 - 4b}}{2} \]
For both roots to be in the range \([0, 1]\), we analyze \(y_1\) and \(y_2\).

1. **For \(y_1 \geq 0\)**:
\[ \frac{-1 + \sqrt{1 - 4b}}{2} \geq 0 \quad \Rightarrow \quad -1 + \sqrt{1 - 4b} \geq 0 \quad \Rightarrow \quad \sqrt{1 - 4b} \geq 1 \]
Squaring both sides gives:
\[ 1 - 4b \geq 1 \quad \Rightarrow \quad -4b \geq 0 \quad \Rightarrow \quad b \leq 0 \]

2. **For \(y_2 < 1\)**:
\[ \frac{-1 - \sqrt{1 - 4b}}{2} < 1 \quad \Rightarrow \quad -1 - \sqrt{1 - 4b} < 2 \quad \Rightarrow \quad -\sqrt{1 - 4b} < 3 \]
This is always true since \(\sqrt{1 - 4b} \geq 0\).

### Step 6: Combine the conditions
From the conditions derived, we have:
- \(b < \frac{1}{4}\)
- \(b \leq 0\)

The more restrictive condition is \(b \leq 0\).

### Step 7: Determine the final range
Since we also need to ensure that the roots are distinct, we also need \(1 - 4b > 0\), which gives:
\[ b < \frac{1}{4} \]
However, since \(b\) must also be less than or equal to 0, we conclude:
\[ -2 < b < 0 \]

### Conclusion
Thus, the values of \(b\) for which both distinct roots of the equation are real and lie in the interval \([0, \pi]\) are:
\[ b \in (-2, 0) \]