The number of values of x in the interval `[0, 3pi]` satisfying the equation `2sin^2x + 5sin x- 3 = 0` is

Options

A. 6
B. 1
C. 2
D. 4

Answer

To solve the equation \(2\sin^2 x + 5\sin x - 3 = 0\) for the number of values of \(x\) in the interval \([0, 3\pi]\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the quadratic equation in terms of \(\sin x\):
\[ 2\sin^2 x + 5\sin x - 3 = 0 \]

### Step 2: Factor the quadratic equation
We will factor the quadratic equation. We need to find two numbers that multiply to \(2 \times (-3) = -6\) and add up to \(5\). The numbers \(6\) and \(-1\) work:
\[ 2\sin^2 x + 6\sin x - \sin x - 3 = 0 \]
Now, we can group the terms:
\[ (2\sin^2 x + 6\sin x) + (-\sin x - 3) = 0 \]
Factoring by grouping gives us:
\[ 2\sin x(\sin x + 3) - 1(\sin x + 3) = 0 \]
This can be factored further:
\[ (2\sin x - 1)(\sin x + 3) = 0 \]

### Step 3: Solve for \(\sin x\)
Now we set each factor to zero:
1. \(2\sin x - 1 = 0\)
2. \(\sin x + 3 = 0\)

From the first equation:
\[ 2\sin x = 1 \implies \sin x = \frac{1}{2} \]

From the second equation:
\[ \sin x = -3 \]
This value is not possible since the sine function only takes values in the range \([-1, 1]\).

### Step 4: Find the angles for \(\sin x = \frac{1}{2}\)
The angles that satisfy \(\sin x = \frac{1}{2}\) are:
\[ x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad x = \frac{5\pi}{6} + 2k\pi \quad \text{for integer } k \]

### Step 5: Find the values in the interval \([0, 3\pi]\)
We will find the values of \(x\) for \(k = 0\) and \(k = 1\):

1. For \(k = 0\):
- \(x = \frac{\pi}{6}\)
- \(x = \frac{5\pi}{6}\)

2. For \(k = 1\):
- \(x = \frac{\pi}{6} + 2\pi = \frac{13\pi}{6}\)
- \(x = \frac{5\pi}{6} + 2\pi = \frac{17\pi}{6}\)

### Step 6: List all valid solutions
The valid solutions in the interval \([0, 3\pi]\) are:
- \(x = \frac{\pi}{6}\)
- \(x = \frac{5\pi}{6}\)
- \(x = \frac{13\pi}{6}\)
- \(x = \frac{17\pi}{6}\)

### Conclusion
Thus, the total number of values of \(x\) that satisfy the original equation in the interval \([0, 3\pi]\) is **4**.