1. **State the problem:** Simplify or analyze the quadratic expression $x^2 - 13x - 5$. 2. **Formula and rules:** The quadratic expression is in the form $ax^2 + bx + c$ where $a=1$, $b=-13$, and $c=-5$. 3. **Find the roots using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Calculate the discriminant:** $$b^2 - 4ac = (-13)^2 - 4 \times 1 \times (-5) = 169 + 20 = 189$$ 5. **Substitute values into the formula:** $$x = \frac{-(-13) \pm \sqrt{189}}{2 \times 1} = \frac{13 \pm \sqrt{189}}{2}$$ 6. **Simplify the square root if possible:** $$\sqrt{189} = \sqrt{9 \times 21} = 3\sqrt{21}$$ 7. **Final roots:** $$x = \frac{13 \pm 3\sqrt{21}}{2}$$ This gives the two solutions for $x$. **Answer:** The roots of the quadratic $x^2 - 13x - 5$ are $$x = \frac{13 + 3\sqrt{21}}{2}$$ and $$x = \frac{13 - 3\sqrt{21}}{2}$$.