The number of solutions of the equation `sin((pix)/(2sqrt3))=x^2-2sqrt3x+4`

Options

A. 0
B. 2
C. more than 2
D. 1

Answer

To find the number of solutions of the equation

\[ \sin\left(\frac{\pi x}{2\sqrt{3}}\right) = x^2 - 2\sqrt{3}x + 4, \]

we will analyze both sides of the equation step by step.

### Step 1: Define the functions
Let:
- \( f(x) = \sin\left(\frac{\pi x}{2\sqrt{3}}\right) \)
- \( g(x) = x^2 - 2\sqrt{3}x + 4 \)

### Step 2: Determine the range of \( g(x) \)
The function \( g(x) \) is a quadratic function. To find its range, we need to calculate its vertex.

The vertex \( x \) of a quadratic \( ax^2 + bx + c \) is given by:

\[ x = -\frac{b}{2a} \]

Here, \( a = 1 \) and \( b = -2\sqrt{3} \).

Calculating the vertex:

\[ x = -\frac{-2\sqrt{3}}{2 \cdot 1} = \sqrt{3} \]

Now, we will find the value of \( g(\sqrt{3}) \):

\[ g(\sqrt{3}) = (\sqrt{3})^2 - 2\sqrt{3}(\sqrt{3}) + 4 = 3 - 6 + 4 = 1 \]

Since the parabola opens upwards (as \( a > 0 \)), the minimum value of \( g(x) \) is 1, and it approaches infinity as \( x \) moves away from \( \sqrt{3} \). Thus, the range of \( g(x) \) is:

\[ [1, \infty) \]

### Step 3: Determine the range of \( f(x) \)
The function \( f(x) = \sin\left(\frac{\pi x}{2\sqrt{3}}\right) \) oscillates between -1 and 1 for all \( x \). Therefore, the range of \( f(x) \) is:

\[ [-1, 1] \]

### Step 4: Analyze the intersection of ranges
Now we compare the ranges of \( f(x) \) and \( g(x) \):
- Range of \( f(x) \): \([-1, 1]\)
- Range of \( g(x) \): \([1, \infty)\)

The only point where these ranges overlap is at \( y = 1 \).

### Step 5: Find the number of solutions
To find the number of solutions to the equation \( f(x) = g(x) \), we need to check if there is a solution at the point where \( g(x) = 1 \).

Setting \( g(x) = 1 \):

\[ x^2 - 2\sqrt{3}x + 4 = 1 \]

This simplifies to:

\[ x^2 - 2\sqrt{3}x + 3 = 0 \]

Calculating the discriminant \( D \):

\[ D = b^2 - 4ac = (-2\sqrt{3})^2 - 4 \cdot 1 \cdot 3 = 12 - 12 = 0 \]

Since the discriminant is zero, there is exactly one solution for \( g(x) = 1 \).

### Conclusion
Since \( f(x) \) can equal \( g(x) \) only at \( y = 1 \) and there is exactly one \( x \) such that \( g(x) = 1 \), we conclude that there is exactly one solution to the equation.

Thus, the number of solutions of the equation is:

\[ \boxed{1} \]