To express the solution of the quadratic equation \( x^2 + 2x + 8 = 0 \) in the form \( a + bi \), where \( i \) is the imaginary unit, follow these steps:

### Step 1: Identify coefficients
The equation is in the form \( ax^2 + bx + c = 0 \). For this equation:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 8 \)

### Step 2: Apply the quadratic formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substitute \( a = 1 \), \( b = 2 \), and \( c = 8 \) into the formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(8)}}{2(1)}
\]
\[
x = \frac{-2 \pm \sqrt{4 - 32}}{2}
\]
\[
x = \frac{-2 \pm \sqrt{-28}}{2}
\]

### Step 3: Simplify the square root
Since \( \sqrt{-28} = \sqrt{28}i \), we simplify further:
\[
x = \frac{-2 \pm \sqrt{28}i}{2}
\]
\[
x = \frac{-2 \pm 2\sqrt{7}i}{2}
\]

### Step 4: Simplify the expression
Divide both terms by 2:
\[
x = -1 \pm \sqrt{7}i
\]

### Final Answer:
The solutions are:
\[
x = -1 + \sqrt{7}i \quad \text{and} \quad x = -1 - \sqrt{7}i
\]
Thus, the solutions are expressed in the form \( a + bi \).

Would you like more details or have any questions about this solution? Here are some follow-up questions for you:

1. What is the discriminant and how does it determine the nature of the roots?
2. Can you solve a similar quadratic equation with different coefficients?
3. How would the solution change if the constant term were negative?
4. What is the significance of the imaginary unit \( i \)?
5. How can you graph the solutions on the complex plane?

**Tip:** The discriminant (\( b^2 - 4ac \)) determines whether the solutions are real or complex. If it's negative, the solutions will involve imaginary numbers.

Learn how to solve the quadratic equation x^2 + 2x + 8 = 0 and express the solution in the form a + bi. This problem involves key concepts like the quadratic formula and complex numbers, suitable for students in Grades 9-11. Detailed step-by-step solution provided.

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