1. **State the problem:** Solve the quadratic equation $4x^2 - 1 = 6x$ using the quadratic formula. 2. **Rewrite the equation in standard form:** Move all terms to one side: $$4x^2 - 6x - 1 = 0$$ 3. **Identify coefficients:** Here, $a=4$, $b=-6$, and $c=-1$. 4. **Recall the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-6)^2 - 4 \times 4 \times (-1) = 36 + 16 = 52$$ 6. **Find the square root of the discriminant:** $$\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$$ 7. **Substitute values into the formula:** $$x = \frac{-(-6) \pm 2\sqrt{13}}{2 \times 4} = \frac{6 \pm 2\sqrt{13}}{8}$$ 8. **Simplify the expression:** $$x = \frac{6}{8} \pm \frac{2\sqrt{13}}{8} = \frac{3}{4} \pm \frac{\sqrt{13}}{4}$$ **Final answer:** $$x = \frac{3 \pm \sqrt{13}}{4}$$