1. **State the problem:** Solve the equation $2(2x - 3) - (6x - 9) = 2x^2$ for $x$. 2. **Apply the distributive property:** $$2(2x - 3) = 4x - 6$$ $$-(6x - 9) = -6x + 9$$ So the left side becomes: $$4x - 6 - 6x + 9$$ 3. **Combine like terms on the left side:** $$4x - 6x = -2x$$ $$-6 + 9 = 3$$ So the equation is: $$-2x + 3 = 2x^2$$ 4. **Rewrite the equation to standard quadratic form:** $$2x^2 + 2x - 3 = 0$$ 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=2$, $b=2$, $c=-3$. 6. **Calculate the discriminant:** $$b^2 - 4ac = 2^2 - 4(2)(-3) = 4 + 24 = 28$$ 7. **Find the roots:** $$x = \frac{-2 \pm \sqrt{28}}{2 \times 2} = \frac{-2 \pm 2\sqrt{7}}{4}$$ 8. **Simplify the fraction:** $$x = \frac{\cancel{-2} \pm \cancel{2}\sqrt{7}}{\cancel{4}} = \frac{-1 \pm \sqrt{7}}{2}$$ 9. **Final answer:** $$x = \frac{-1 + \sqrt{7}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{7}}{2}$$