1. **State the problem:** Solve the quadratic equation $$2x^2 - 19 = 0$$ by completing the square. 2. **Rewrite the equation:** Move the constant term to the right side: $$2x^2 = 19$$ 3. **Divide both sides by 2 to isolate $x^2$:** $$x^2 = \frac{19}{2}$$ 4. **Complete the square:** Since the equation is already a perfect square form of $x^2$, we can write it as: $$(x + 0)^2 = \frac{19}{2}$$ 5. **Solve for $x$ by taking the square root of both sides:** $$x + 0 = \pm \sqrt{\frac{19}{2}}$$ 6. **Simplify the square root:** $$x = \pm \frac{\sqrt{19}}{\sqrt{2}} = \pm \frac{\sqrt{19} \sqrt{2}}{2} = \pm \frac{\sqrt{38}}{2}$$ **Final answers:** $$x = \frac{\sqrt{38}}{2}, \quad x = -\frac{\sqrt{38}}{2}$$