To solve the quadratic equation \( 2x^2 + 7x + 9 = 0 \), you can use the **quadratic formula**, which is:

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

For the given equation \( 2x^2 + 7x + 9 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 7 \)
- \( c = 9 \)

### Step-by-Step Solution:

1. **Calculate the discriminant**:
   \[
   \Delta = b^2 - 4ac = 7^2 - 4(2)(9) = 49 - 72 = -23
   \]
   Since the discriminant \( \Delta = -23 \) is negative, the equation has no real solutions, but it does have two complex solutions.

2. **Apply the quadratic formula**:
   \[
   x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-7 \pm \sqrt{-23}}{2(2)} = \frac{-7 \pm \sqrt{23}i}{4}
   \]
   Therefore, the solutions are:
   \[
   x_1 = \frac{-7 + \sqrt{23}i}{4} \quad \text{and} \quad x_2 = \frac{-7 - \sqrt{23}i}{4}
   \]

These are the two complex roots of the quadratic equation.

Would you like more details or have any further questions?

### 5 related questions:
1. What are the possible types of solutions for a quadratic equation?
2. How does the discriminant determine the nature of the solutions?
3. Can quadratic equations always be solved by factoring?
4. What is the geometric interpretation of complex roots?
5. How do imaginary numbers arise in quadratic equations?

### Tip:
Always check the discriminant first when solving a quadratic equation to determine whether the solutions are real or complex.

Solve the quadratic equation 2x² + 7x + 9 = 0 using the quadratic formula. This step-by-step guide explains how to calculate the discriminant and find complex solutions. Suitable for students in Grades 9-10.

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