Solve the following quadratic: `x^2+2x+2=0`

Explanation

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

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

### Step 1: Identify coefficients
In the equation \(x^2 + 2x + 2 = 0\), we identify the coefficients:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 2\) (coefficient of \(x\))
- \(c = 2\) (constant term)

### Step 2: Calculate the discriminant
Next, we calculate the discriminant \(D = b^2 - 4ac\):
\[ D = 2^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4 \]

### Step 3: Apply the quadratic formula
Since the discriminant is negative, we will have complex roots. We substitute \(a\), \(b\), and \(D\) into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{-4}}{2 \cdot 1} \]

### Step 4: Simplify the square root
The square root of \(-4\) can be simplified using the imaginary unit \(i\):
\[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]

### Step 5: Substitute back into the formula
Now, we substitute \(\sqrt{-4}\) back into the formula:
\[ x = \frac{-2 \pm 2i}{2} \]

### Step 6: Simplify the expression
We can simplify this expression:
\[ x = \frac{-2}{2} \pm \frac{2i}{2} = -1 \pm i \]

### Step 7: Write the final solutions
Thus, the solutions to the quadratic equation \(x^2 + 2x + 2 = 0\) are:
\[ x = -1 + i \quad \text{and} \quad x = -1 - i \]

### Summary of solutions:
The roots of the equation are:
\[ x_1 = -1 + i, \quad x_2 = -1 - i \]

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