The expression you provided is a quadratic equation:

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

To solve this quadratic equation, we can use the quadratic formula:

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

Where:
- \(a = 1\) (coefficient of \(x^2\)),
- \(b = 5\) (coefficient of \(x\)),
- \(c = -3\) (constant term).

Let's calculate the discriminant (\(b^2 - 4ac\)) first:

\[
\text{Discriminant} = 5^2 - 4(1)(-3) = 25 + 12 = 37
\]

Since the discriminant is positive, there are two real solutions.

Now, we substitute into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{37}}{2(1)}
\]
\[
x = \frac{-5 \pm \sqrt{37}}{2}
\]

So, the solutions are:

\[
x_1 = \frac{-5 + \sqrt{37}}{2}, \quad x_2 = \frac{-5 - \sqrt{37}}{2}
\]

These are the exact solutions. If you want approximate values, you can calculate:

\[
x_1 \approx \frac{-5 + 6.08}{2} = \frac{1.08}{2} \approx 0.54
\]
\[
x_2 \approx \frac{-5 - 6.08}{2} = \frac{-11.08}{2} \approx -5.54
\]

Thus, the approximate solutions are:

\[
x_1 \approx 0.54, \quad x_2 \approx -5.54
\]

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

Here are some related questions:
1. How do we interpret the discriminant in a quadratic equation?
2. Can you graph the quadratic function \(x^2 + 5x - 3\)?
3. How would the solutions change if the discriminant were negative?
4. What are the different methods to solve a quadratic equation?
5. How does completing the square compare to using the quadratic formula?

**Tip:** The discriminant in a quadratic equation tells you the nature of the roots—positive for real roots, zero for a repeated root, and negative for complex roots.

Learn how to solve the quadratic equation x^2 + 5x - 3 using the quadratic formula. This detailed guide explains the steps, including calculating the discriminant and finding the real roots. Suitable for students in Grades 9-11.

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