1. **State the problem:** Solve the quadratic equation $4x^2 - 1 = 0$ for $x$. 2. **Formula and rules:** This is a quadratic equation in the form $ax^2 + bx + c = 0$. Here, $a=4$, $b=0$, and $c=-1$. We can solve it by isolating $x^2$ and then taking the square root. 3. **Isolate $x^2$:** $$4x^2 - 1 = 0$$ Add 1 to both sides: $$4x^2 = 1$$ 4. **Divide both sides by 4:** $$\frac{4x^2}{\cancel{4}} = \frac{1}{\cancel{4}}$$ $$x^2 = \frac{1}{4}$$ 5. **Take the square root of both sides:** $$x = \pm \sqrt{\frac{1}{4}}$$ $$x = \pm \frac{1}{2}$$ 6. **Final answer:** $$x = \frac{1}{2} \quad \text{or} \quad x = -\frac{1}{2}$$ These are the two solutions to the equation.