We can solve the quadratic equation \( x^2 + 8x + 9 = 0 \) using either factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

The quadratic formula is:

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

For the equation \( x^2 + 8x + 9 = 0 \), the coefficients are:

\[
a = 1, \quad b = 8, \quad c = 9
\]

### Step 1: Calculate the discriminant (\( \Delta \)):

\[
\Delta = b^2 - 4ac = 8^2 - 4(1)(9) = 64 - 36 = 28
\]

### Step 2: Plug into the quadratic formula:

\[
x = \frac{-8 \pm \sqrt{28}}{2(1)} = \frac{-8 \pm \sqrt{28}}{2}
\]

### Step 3: Simplify:

Since \( \sqrt{28} = 2\sqrt{7} \), we get:

\[
x = \frac{-8 \pm 2\sqrt{7}}{2} = -4 \pm \sqrt{7}
\]

Thus, the solutions are:

\[
x = -4 + \sqrt{7} \quad \text{or} \quad x = -4 - \sqrt{7}
\]

These are the two roots of the equation.

Would you like more details on any step or have questions about the solution? 

Here are 5 related questions to expand on this topic:
1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. How can you solve this equation by completing the square?
4. What are real-world applications of quadratic equations?
5. How would the solution change if the discriminant were negative?

**Tip:** Always check the discriminant first—it helps you quickly understand the nature of the roots (real or complex).

Learn how to solve the quadratic equation x^2 + 8x + 9 = 0 using the quadratic formula. This step-by-step guide explains the calculation of the discriminant and root simplification. Suitable for students in Grades 9-10.

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