Without solving equation `x^(2)-x+1=0`, find whether `x=-1` is a root of this equation or root.

Answer

To determine whether \( x = -1 \) is a root of the equation \( x^2 - x + 1 = 0 \) without solving the equation, we can substitute \( x = -1 \) into the left-hand side of the equation and check if it equals the right-hand side, which is 0.

### Step-by-step Solution:

1. **Substitute \( x = -1 \) into the equation**:
\[ x^2 - x + 1 \]
becomes:
\[ (-1)^2 - (-1) + 1 \]

2. **Calculate \( (-1)^2 \)**:
\[ (-1)^2 = 1 \]

3. **Calculate \( -(-1) \)**:
\[ -(-1) = 1 \]

4. **Combine the results**:
Now substituting back into the expression:
\[ 1 + 1 + 1 = 3 \]

5. **Compare with the right-hand side**:
The right-hand side of the equation is 0. We found that the left-hand side equals 3:
\[ 3 \neq 0 \]

6. **Conclusion**:
Since the left-hand side does not equal the right-hand side, we conclude that \( x = -1 \) is not a root of the equation \( x^2 - x + 1 = 0 \).

### Final Answer:
\( x = -1 \) is not a root of the equation \( x^2 - x + 1 = 0 \).

---