1. **State the problem:** Solve the quadratic equation $4x^2 - 3 = 12x$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$4x^2 - 12x - 3 = 0$$ 3. **Identify coefficients:** Here, $a=4$, $b=-12$, and $c=-3$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = (-12)^2 - 4 \times 4 \times (-3) = 144 + 48 = 192$$ 6. **Simplify the square root:** $$\sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3}$$ 7. **Substitute values into the formula:** $$x = \frac{-(-12) \pm 8\sqrt{3}}{2 \times 4} = \frac{12 \pm 8\sqrt{3}}{8}$$ 8. **Simplify the fraction:** $$x = \frac{12}{8} \pm \frac{8\sqrt{3}}{8} = \frac{3}{2} \pm \sqrt{3}$$ **Final answer:** $$x = \frac{3}{2} + \sqrt{3} \quad \text{or} \quad x = \frac{3}{2} - \sqrt{3}$$