1. **State the problem:** Solve the equation $$\frac{1}{2} x^2 = 128$$ for $x$. 2. **Formula and rules:** This is a quadratic equation in the form $$ax^2 = b$$ where $a = \frac{1}{2}$ and $b = 128$. To solve for $x$, multiply both sides by the reciprocal of $a$ and then take the square root. 3. **Intermediate work:** Multiply both sides by 2: $$x^2 = 128 \times 2 = 256$$ 4. **Take the square root of both sides:** $$x = \pm \sqrt{256}$$ 5. **Evaluate the square root:** $$x = \pm 16$$ 6. **Final answer:** The solutions to the equation are $$x = 16$$ and $$x = -16$$. This means the parabola described by $\frac{1}{2} x^2$ equals 128 at these two points.