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The image contains two quadratic equations:

1. \( x^2 + 3x + 9 = 0 \)
2. \( -2x^2 - 11x - 40 = 0 \)

### Solution 1: \( x^2 + 3x + 9 = 0 \)

This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \).

To find the roots, we use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 3 \), and \( c = 9 \).

Substituting these values:
\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(9)}}{2(1)}
\]
\[
x = \frac{-3 \pm \sqrt{9 - 36}}{2}
\]
\[
x = \frac{-3 \pm \sqrt{-27}}{2}
\]
\[
x = \frac{-3 \pm \sqrt{27i^2}}{2}
\]
\[
x = \frac{-3 \pm 3\sqrt{3}i}{2}
\]
Thus, the solutions are:
\[
x = \frac{-3 + 3\sqrt{3}i}{2} \quad \text{and} \quad x = \frac{-3 - 3\sqrt{3}i}{2}
\]
These are complex roots.

### Solution 2: \( -2x^2 - 11x - 40 = 0 \)

This is also a quadratic equation. Here, \( a = -2 \), \( b = -11 \), and \( c = -40 \).

Applying the quadratic formula:
\[
x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(-2)(-40)}}{2(-2)}
\]
\[
x = \frac{11 \pm \sqrt{121 - 320}}{-4}
\]
\[
x = \frac{11 \pm \sqrt{-199}}{-4}
\]
\[
x = \frac{11 \pm \sqrt{199i^2}}{-4}
\]
\[
x = \frac{11 \pm \sqrt{199}i}{-4}
\]
Thus, the solutions are:
\[
x = \frac{-11 + \sqrt{199}i}{4} \quad \text{and} \quad x = \frac{-11 - \sqrt{199}i}{4}
\]
These are also complex roots.

Would you like more details on these solutions or have any specific questions? Here are some related questions:

1. How can we derive the quadratic formula?
2. What is the significance of the discriminant in determining the nature of roots?
3. How do we interpret complex roots geometrically?
4. Can a quadratic equation with real coefficients have only one complex root?
5. What are the implications of having a negative discriminant in a quadratic equation?

**Tip:** When solving quadratic equations, the discriminant \( b^2 - 4ac \) can quickly tell you if the roots are real or complex without needing to compute the entire quadratic formula.

Solve the quadratic equations x^2 + 3x + 9 = 0 and -2x^2 - 11x - 40 = 0.

Learn how to solve the quadratic equations x^2 + 3x + 9 = 0 and -2x^2 - 11x - 40 = 0 using the quadratic formula. Discover the steps to find complex roots when the discriminant is negative. Ideal for students in Grades 9-12.

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