The equation you've provided is:

\[
x^2 + 3x + 5 = 0
\]

This is a quadratic equation, and it can be solved using the quadratic formula. The quadratic formula is:

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

For the equation \(x^2 + 3x + 5 = 0\), the coefficients are:
- \( a = 1 \)
- \( b = 3 \)
- \( c = 5 \)

Let's substitute these values into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(5)}}{2(1)}
\]

Simplifying this:

\[
x = \frac{-3 \pm \sqrt{9 - 20}}{2}
\]

\[
x = \frac{-3 \pm \sqrt{-11}}{2}
\]

Since the discriminant (\(\sqrt{-11}\)) is negative, the solutions are complex numbers. We can express the square root of \(-11\) as \( \sqrt{11}i \), where \( i \) is the imaginary unit.

Thus, the solutions are:

\[
x = \frac{-3 + \sqrt{11}i}{2}
\]
and
\[
x = \frac{-3 - \sqrt{11}i}{2}
\]

In summary, the solutions are:

\[
x = -\frac{3}{2} + \frac{\sqrt{11}}{2}i
\]
and
\[
x = -\frac{3}{2} - \frac{\sqrt{11}}{2}i
\]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What are complex numbers and how are they used in mathematics?
2. How do you determine if a quadratic equation has real or complex solutions?
3. What is the significance of the discriminant in the quadratic formula?
4. How do you graph a quadratic equation with complex roots?
5. Can a quadratic equation have only one solution? If so, when?

**Tip:** If the discriminant (\(b^2 - 4ac\)) is negative, the quadratic equation has complex roots, not real ones.

Explore how to solve the quadratic equation Xsquare+3x+5=0 with complex number solutions. Understand the quadratic formula, discriminant, and complex roots in algebra. Suitable for Grades 10-12.

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