To solve the quadratic equation \( x^2 + 5x + 3 = 0 \), we can use the quadratic formula:

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

For the given equation \( x^2 + 5x + 3 = 0 \):

- \( a = 1 \)
- \( b = 5 \)
- \( c = 3 \)

Substitute these values into the quadratic formula:

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

\[
x = \frac{{-5 \pm \sqrt{{25 - 12}}}}{2}
\]

\[
x = \frac{{-5 \pm \sqrt{13}}}{2}
\]

Thus, the solutions are:

\[
x = \frac{{-5 + \sqrt{13}}}{2} \quad \text{and} \quad x = \frac{{-5 - \sqrt{13}}}{2}
\]

These are the two roots of the equation. Would you like further details on this solution or have any other questions?

### Related Questions
1. How do you derive the quadratic formula?
2. What is the discriminant and how does it determine the nature of roots?
3. Can you graph the equation \( x^2 + 5x + 3 = 0 \)?
4. How do you complete the square for the equation \( x^2 + 5x + 3 = 0 \)?
5. What are some real-world applications of solving quadratic equations?

### Tip
For any quadratic equation, the discriminant (\( b^2 - 4ac \)) helps determine if the roots are real, repeated, or complex.

Learn how to solve the quadratic equation x^2 + 5x + 3 = 0 with a step-by-step solution using the quadratic formula. This example demonstrates key algebraic concepts such as solving quadratic equations and calculating the discriminant, suitable for Grades 8-10.

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