Find the nature of the roots of `x^(2)-x+1=0`

Answer

To find the nature of the roots of the quadratic equation \( x^2 - x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 1 \)

### Step 2: Calculate the discriminant
The nature of the roots of a quadratic equation can be determined using the discriminant \( D \), which is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 \]
\[ D = 1 - 4 \]
\[ D = -3 \]

### Step 3: Determine the nature of the roots
The nature of the roots is determined based on the value of the discriminant \( D \):
- If \( D > 0 \), the roots are real and distinct.
- If \( D = 0 \), the roots are real and equal.
- If \( D < 0 \), the roots are imaginary (complex).

Since we found that \( D = -3 \), which is less than 0, we conclude that the roots of the equation are imaginary.

### Final Answer
The roots of the equation \( x^2 - x + 1 = 0 \) are imaginary.

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