If ` sqrt(3) ( cos^2 x) = ( sqrt(3)-1) cos x +1,` the numbers of solution of the given equation when ` x in [0,pi/2]` is

Answer

To solve the equation \( \sqrt{3} \cos^2 x = (\sqrt{3} - 1) \cos x + 1 \) for \( x \) in the interval \( [0, \frac{\pi}{2}] \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to bring all terms to one side:

\[ \sqrt{3} \cos^2 x - (\sqrt{3} - 1) \cos x - 1 = 0 \]

### Step 2: Substituting \( t = \cos x \)
Let \( t = \cos x \). The equation then becomes:

\[ \sqrt{3} t^2 - (\sqrt{3} - 1) t - 1 = 0 \]

### Step 3: Identifying the Quadratic Equation
This is a quadratic equation in terms of \( t \):

\[ \sqrt{3} t^2 - (\sqrt{3} - 1) t - 1 = 0 \]

### Step 4: Using the Quadratic Formula
We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( t \):

Here, \( a = \sqrt{3} \), \( b = -(\sqrt{3} - 1) \), and \( c = -1 \).

Calculating the discriminant:

\[ D = b^2 - 4ac = [-(\sqrt{3} - 1)]^2 - 4(\sqrt{3})(-1) \]

Calculating \( D \):

\[ D = (\sqrt{3} - 1)^2 + 4\sqrt{3} = 3 - 2\sqrt{3} + 1 + 4\sqrt{3} = 4 + 2\sqrt{3} \]

### Step 5: Finding the Roots
Now we can find the roots using the quadratic formula:

\[ t = \frac{(\sqrt{3} - 1) \pm \sqrt{4 + 2\sqrt{3}}}{2\sqrt{3}} \]

### Step 6: Evaluating the Roots
We need to check if the roots \( t \) lie within the range \( [0, 1] \) since \( t = \cos x \) and \( x \) is in \( [0, \frac{\pi}{2}] \).

### Step 7: Analyzing the Roots
1. **Root 1**: \( t_1 = \frac{(\sqrt{3} - 1) + \sqrt{4 + 2\sqrt{3}}}{2\sqrt{3}} \)
2. **Root 2**: \( t_2 = \frac{(\sqrt{3} - 1) - \sqrt{4 + 2\sqrt{3}}}{2\sqrt{3}} \)

We need to determine if these roots are within the interval [0, 1].

### Step 8: Conclusion
After evaluating the roots, we find that:
- \( t_1 \) is positive and less than or equal to 1.
- \( t_2 \) is negative, which is not in the range of \( \cos x \).

Thus, there is **one valid solution** for \( x \) in the interval \( [0, \frac{\pi}{2}] \).

### Final Answer
The number of solutions of the given equation when \( x \) is in \( [0, \frac{\pi}{2}] \) is **1**.

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