1. **State the problem:** Solve the quadratic equation $$3x^2 = 4 - 4x$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$3x^2 + 4x - 4 = 0$$ 3. **Identify coefficients:** Here, $a = 3$, $b = 4$, and $c = -4$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = 4^2 - 4 \times 3 \times (-4) = 16 + 48 = 64$$ 6. **Calculate the roots:** $$x = \frac{-4 \pm \sqrt{64}}{2 \times 3} = \frac{-4 \pm 8}{6}$$ 7. **Find each solution:** - For the plus sign: $$x = \frac{-4 + 8}{6} = \frac{4}{6} = \frac{2}{3}$$ - For the minus sign: $$x = \frac{-4 - 8}{6} = \frac{-12}{6} = -2$$ 8. **Final answer:** $$x = \frac{2}{3}, \quad x = -2$$