1. **State the problem:** Solve the quadratic equation $$x^2 + 8x = 13$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 + 8x - 13 = 0$$ 3. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=8$, and $c=-13$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 8^2 - 4 \times 1 \times (-13) = 64 + 52 = 116$$ 5. **Substitute values into the quadratic formula:** $$x = \frac{-8 \pm \sqrt{116}}{2 \times 1} = \frac{-8 \pm \sqrt{116}}{2}$$ 6. **Simplify the square root:** $$\sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29}$$ 7. **Simplify the fraction:** $$x = \frac{-8 \pm 2\sqrt{29}}{2}$$ 8. **Cancel common factor 2 in numerator and denominator:** $$x = \frac{\cancel{2}(-4 \pm \sqrt{29})}{\cancel{2}} = -4 \pm \sqrt{29}$$ 9. **Final solutions:** $$x = -4 + \sqrt{29} \quad \text{or} \quad x = -4 - \sqrt{29}$$