The given equation is a quadratic equation:

$x^2 + x + 1 = 0$

Step 1: General Form of Quadratic Equation

A quadratic equation is written in the standard form: $ax^2 + bx + c = 0$ Here, $a = 1$, $b = 1$, and $c = 1$.

Step 2: Solve Using the Quadratic Formula

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting $a = 1$, $b = 1$, and $c = 1$ into the formula: $x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)}$

Step 3: Simplify the Discriminant

$\text{Discriminant} = b^2 - 4ac = 1 - 4 = -3$

Since the discriminant is negative ($-3$), the roots are complex.

Step 4: Express the Roots

$x = \frac{-1 \pm \sqrt{-3}}{2}$

We simplify further: $x = \frac{-1 \pm i\sqrt{3}}{2}$

Final Answer:

The roots are: $x = \frac{-1 + i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{-1 - i\sqrt{3}}{2}$

Explanation:

The solutions are complex conjugates because the discriminant is negative. The imaginary part arises from the square root of a negative number ($\sqrt{-3} = i\sqrt{3}$).

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Related Questions:

1. What is the significance of the discriminant in determining root types?
2. How can you represent complex numbers geometrically?
3. Can quadratic equations with real coefficients have only one root?
4. What happens when the discriminant equals zero?
5. How does the quadratic formula relate to completing the square?

Tip:

When solving a quadratic equation, always check the discriminant first to determine the nature of the roots!