1. **State the problem:** Find the roots of the quadratic equation $$2x^2 - 12 - 7 = 0$$. 2. **Rewrite the equation:** Combine like terms: $$2x^2 - 19 = 0$$ 3. **Isolate the quadratic term:** $$2x^2 = 19$$ 4. **Divide both sides by 2:** $$\cancel{2}x^2 = \cancel{2} \times \frac{19}{2}$$ $$x^2 = \frac{19}{2}$$ 5. **Take the square root of both sides:** $$x = \pm \sqrt{\frac{19}{2}}$$ 6. **Simplify the square root:** $$x = \pm \frac{\sqrt{19}}{\sqrt{2}} = \pm \frac{\sqrt{19}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \pm \frac{\sqrt{38}}{2}$$ **Final answer:** $$x = \pm \frac{\sqrt{38}}{2}$$