To solve the quadratic equation \( x^2 + x + 1 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the given equation \( x^2 + x + 1 = 0 \), the coefficients are:
\[ a = 1 \]
\[ b = 1 \]
\[ c = 1 \]

Plugging these values into the quadratic formula:

\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} \]
\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]
\[ x = \frac{-1 \pm \sqrt{3}i}{2} \]

So the solutions are:
\[ x = \frac{-1 + \sqrt{3}i}{2} \]
\[ x = \frac{-1 - \sqrt{3}i}{2} \]

These solutions are complex numbers since the discriminant (\(b^2 - 4ac = -3\)) is negative.

Would you like further details or have any questions?

Here are some related questions you might be interested in:

1. What are the general steps to solve a quadratic equation?
2. How do we determine if the solutions to a quadratic equation are real or complex?
3. What is the significance of the discriminant in a quadratic equation?
4. How can complex solutions be represented on the complex plane?
5. Can you explain the quadratic formula derivation?
6. How can quadratic equations be solved using factoring?
7. What is the role of the imaginary unit \(i\) in complex numbers?
8. How do quadratic equations relate to parabolas?

**Tip:** For any quadratic equation \(ax^2 + bx + c = 0\), the nature of the roots can be determined by the discriminant \(b^2 - 4ac\). If it is positive, the roots are real and distinct; if zero, the roots are real and equal; if negative, the roots are complex.

Learn how to solve the quadratic equation x^2 + x + 1 = 0 with complex number solutions using the quadratic formula. This problem explores complex numbers and the discriminant theorem. Suitable for students in Grades 10-12.

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