1. **State the problem:** Solve the quadratic equation $3x^2 - 20x - 13 = -6$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$3x^2 - 20x - 13 + 6 = 0$$ which simplifies to $$3x^2 - 20x - 7 = 0$$ 3. **Identify coefficients:** Here, $a = 3$, $b = -20$, and $c = -7$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = (-20)^2 - 4 \times 3 \times (-7) = 400 + 84 = 484$$ 6. **Find the square root of the discriminant:** $$\sqrt{484} = 22$$ 7. **Substitute values into the quadratic formula:** $$x = \frac{-(-20) \pm 22}{2 \times 3} = \frac{20 \pm 22}{6}$$ 8. **Calculate the two possible solutions:** - For the plus sign: $$x = \frac{20 + 22}{6} = \frac{42}{6} = 7$$ - For the minus sign: $$x = \frac{20 - 22}{6} = \frac{\cancel{20 - 22}}{6} = \frac{-2}{6} = \frac{\cancel{-2}}{\cancel{6}} = -\frac{1}{3}$$ 9. **Final answer:** $$x = 7 \quad \text{or} \quad x = -\frac{1}{3}$$