To solve the quadratic equation:

$x^2 + 3x - 5 = 0$

we can apply the quadratic formula:

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

For the given equation, $a = 1$, $b = 3$, and $c = -5$. Substituting these values into the formula:

$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)}$ $x = \frac{-3 \pm \sqrt{9 + 20}}{2}$ $x = \frac{-3 \pm \sqrt{29}}{2}$

Thus, the solutions are:

$x = \frac{-3 + \sqrt{29}}{2} \quad \text{or} \quad x = \frac{-3 - \sqrt{29}}{2}$

These are the exact roots of the equation.

Would you like to explore the decimal approximations or any further details?

Here are 5 related questions:

1. What are the properties of quadratic equations?
2. How can you derive the quadratic formula?
3. What is the discriminant, and how does it help classify the roots?
4. Can you solve the quadratic equation by completing the square?
5. How can graphing help visualize the solutions of quadratic equations?

Tip: The discriminant $b^2 - 4ac$ helps you determine the number and type of roots without solving the entire equation.