Solve: `x^(2)+x+1=0`.

Answer

To solve the quadratic equation \(x^2 + x + 1 = 0\), we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). For our equation:
- \(a = 1\)
- \(b = 1\)
- \(c = 1\)

### Step 2: Use the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\) into the formula:
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

### Step 3: Calculate the discriminant
Now, calculate the discriminant \(b^2 - 4ac\):
\[ 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

### Step 4: Substitute the discriminant back into the formula
Now substitute the discriminant back into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]

### Step 5: Simplify the square root of the negative number
Since \(\sqrt{-3} = \sqrt{3} \cdot i\) (where \(i\) is the imaginary unit), we can rewrite the equation as:
\[ x = \frac{-1 \pm \sqrt{3}i}{2} \]

### Step 6: Write the final solution
Thus, the solutions to the equation \(x^2 + x + 1 = 0\) are:
\[ x = \frac{-1 + \sqrt{3}i}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{3}i}{2} \]