1. **State the problem:** Find the roots of the quadratic equation $$2x^2 - 12x - 7 = 0$$. 2. **Formula used:** The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 2$$, $$b = -12$$, and $$c = -7$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-12)^2 - 4 \times 2 \times (-7) = 144 + 56 = 200$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-12) \pm \sqrt{200}}{2 \times 2} = \frac{12 \pm \sqrt{200}}{4}$$ 6. **Simplify the square root:** $$\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}$$ 7. **Substitute back:** $$x = \frac{12 \pm 10\sqrt{2}}{4}$$ 8. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{2} \times 6 \pm \cancel{2} \times 5\sqrt{2}}{\cancel{2} \times 2} = \frac{6 \pm 5\sqrt{2}}{2}$$ 9. **Final roots:** $$x_1 = \frac{6 + 5\sqrt{2}}{2}, \quad x_2 = \frac{6 - 5\sqrt{2}}{2}$$ These are the two roots of the quadratic equation.