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.