The number of distinct real roots of the equation `sin pi x=x^(2)-x+(5)/(4)` is

Options

A. 0
B. 1
C. 2
D. 4

Explanation

To solve the equation \( \sin(\pi x) = x^2 - x + \frac{5}{4} \) and find the number of distinct real roots, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin(\pi x) = x^2 - x + \frac{5}{4} \]

### Step 2: Analyze the quadratic expression
The right-hand side \( x^2 - x + \frac{5}{4} \) can be rewritten in a more manageable form. We can complete the square:
\[ x^2 - x + \frac{5}{4} = \left(x - \frac{1}{2}\right)^2 + \frac{1}{4} \]
This shows that the minimum value of \( x^2 - x + \frac{5}{4} \) occurs at \( x = \frac{1}{2} \) and is equal to \( \frac{1}{4} \).

### Step 3: Determine the range of the right-hand side
Since \( \left(x - \frac{1}{2}\right)^2 \) is always non-negative, the minimum value of \( x^2 - x + \frac{5}{4} \) is \( \frac{1}{4} \). Therefore, the right-hand side \( x^2 - x + \frac{5}{4} \) is always greater than or equal to \( \frac{1}{4} \).

### Step 4: Analyze the sine function
The left-hand side \( \sin(\pi x) \) oscillates between -1 and 1. Thus, the maximum value of \( \sin(\pi x) \) is 1, and the minimum value is -1.

### Step 5: Set up inequalities
For the equation \( \sin(\pi x) = x^2 - x + \frac{5}{4} \) to have real roots, the right-hand side must lie within the range of the sine function:
\[ -1 \leq x^2 - x + \frac{5}{4} \leq 1 \]
However, since we established that \( x^2 - x + \frac{5}{4} \geq \frac{1}{4} \), we can analyze the inequalities:
1. \( x^2 - x + \frac{5}{4} \geq -1 \) simplifies to:
\[ x^2 - x + \frac{9}{4} \geq 0 \]
This quadratic has no real roots (as the discriminant \( (-1)^2 - 4 \cdot 1 \cdot \frac{9}{4} < 0 \)), meaning it is always positive.

2. \( x^2 - x + \frac{5}{4} \leq 1 \) simplifies to:
\[ x^2 - x + \frac{1}{4} \leq 0 \]
This quadratic can be factored as:
\[ (x - \frac{1}{2})^2 \leq 0 \]
The only solution is \( x = \frac{1}{2} \).

### Step 6: Conclusion
Since the only point where \( x^2 - x + \frac{5}{4} \) intersects \( \sin(\pi x) \) is at \( x = \frac{1}{2} \), and this is the only point where both sides of the equation can be equal, we conclude that there is exactly one distinct real root.

Thus, the number of distinct real roots of the equation is:
\[ \boxed{1} \]