Solve the equation: `z^(2)+z+1=0`. Where z=x+iy.

Answer

To solve the equation \( z^2 + z + 1 = 0 \) where \( z = x + iy \), we can use the quadratic formula. Here are the steps to find the solutions:

### Step 1: Identify coefficients
The given equation is in the standard form of a quadratic equation \( az^2 + bz + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 1 \)

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

### 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:
\[ z = \frac{-1 \pm \sqrt{-3}}{2} \]

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

### Step 6: Separate the real and imaginary parts
This gives us two solutions:
\[ z = \frac{-1 + i\sqrt{3}}{2} \quad \text{and} \quad z = \frac{-1 - i\sqrt{3}}{2} \]

### Final Solutions
Thus, the solutions to the equation \( z^2 + z + 1 = 0 \) are:
\[ z_1 = \frac{-1 + i\sqrt{3}}{2}, \quad z_2 = \frac{-1 - i\sqrt{3}}{2} \]

---