1. **State the problem:** Solve the quadratic equation $$13x^2 + 10x - 3 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=13$, $b=10$, and $c=-3$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 10^2 - 4 \times 13 \times (-3) = 100 + 156 = 256$$ 4. **Evaluate the square root of the discriminant:** $$\sqrt{256} = 16$$ 5. **Apply the quadratic formula:** $$x = \frac{-10 \pm 16}{2 \times 13} = \frac{-10 \pm 16}{26}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{-10 + 16}{26} = \frac{6}{26} = \frac{3}{13}$$ - For the minus sign: $$x = \frac{-10 - 16}{26} = \frac{-26}{26} = -1$$ 7. **Final answer:** $$x = \frac{3}{13} \quad \text{or} \quad x = -1$$