Find the number of value of x in `[0,5pi]` satisying the equation ` 3 cos^2 x -10 cos x +7=0`

Answer

To solve the equation \(3 \cos^2 x - 10 \cos x + 7 = 0\) and find the number of values of \(x\) in the interval \([0, 5\pi]\), we can follow these steps:

### Step 1: Identify the equation
The given equation is:
\[ 3 \cos^2 x - 10 \cos x + 7 = 0 \]
This is a quadratic equation in terms of \(\cos x\).

### Step 2: Let \(y = \cos x\)
We can rewrite the equation as:
\[ 3y^2 - 10y + 7 = 0 \]

### Step 3: Solve the quadratic equation
To solve for \(y\), we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 3\), \(b = -10\), and \(c = 7\).

Calculating the discriminant:
\[ b^2 - 4ac = (-10)^2 - 4 \cdot 3 \cdot 7 = 100 - 84 = 16 \]

Now applying the quadratic formula:
\[ y = \frac{10 \pm \sqrt{16}}{2 \cdot 3} = \frac{10 \pm 4}{6} \]
Calculating the two possible values for \(y\):
1. \(y_1 = \frac{10 + 4}{6} = \frac{14}{6} = \frac{7}{3}\)
2. \(y_2 = \frac{10 - 4}{6} = \frac{6}{6} = 1\)

### Step 4: Analyze the solutions for \(y\)
The value \(y_1 = \frac{7}{3}\) is not possible since the range of \(\cos x\) is \([-1, 1]\). Therefore, we discard this solution.

The only valid solution is:
\[ y = 1 \quad \Rightarrow \quad \cos x = 1 \]

### Step 5: Find the values of \(x\)
The general solution for \(\cos x = 1\) is:
\[ x = 2n\pi \quad \text{for } n \in \mathbb{Z} \]

### Step 6: Determine the values of \(x\) in the interval \([0, 5\pi]\)
Now we need to find the values of \(n\) such that:
\[ 0 \leq 2n\pi \leq 5\pi \]
Dividing the entire inequality by \(2\pi\):
\[ 0 \leq n \leq \frac{5}{2} \]
The integer values of \(n\) that satisfy this are \(n = 0, 1, 2\).

Calculating the corresponding \(x\) values:
- For \(n = 0\): \(x = 0\)
- For \(n = 1\): \(x = 2\pi\)
- For \(n = 2\): \(x = 4\pi\)

### Conclusion
The values of \(x\) that satisfy the equation in the interval \([0, 5\pi]\) are \(0\), \(2\pi\), and \(4\pi\). Thus, the number of values of \(x\) is:
\[ \boxed{3} \]