1. **State the problem:** Solve the quadratic equation $x^2 + 3x - 5 = 0$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=3$, and $c=-5$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times (-5) = 9 + 20 = 29$$ 4. **Apply the quadratic formula:** $$x = \frac{-3 \pm \sqrt{29}}{2 \times 1} = \frac{-3 \pm \sqrt{29}}{2}$$ 5. **Simplify the expression:** No common factors to cancel, so the solutions are: $$x_1 = \frac{-3 + \sqrt{29}}{2}, \quad x_2 = \frac{-3 - \sqrt{29}}{2}$$ 6. **Interpretation:** These are the two real roots of the quadratic equation.