1. **State the problem:** Solve the quadratic equation $$-6x^2 - 9x + 6 = 0$$. 2. **Rewrite the equation:** To simplify, divide the entire equation by -3 to reduce coefficients. $$\cancel{-3} \times \left(-6x^2 - 9x + 6\right) = \cancel{-3} \times 0$$ This gives: $$2x^2 + 3x - 2 = 0$$ 3. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a=2$$, $$b=3$$, and $$c=-2$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$$ 5. **Find the roots:** $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ 6. **Calculate each root:** - For the plus sign: $$x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$$ - For the minus sign: $$x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ 7. **Final answer:** The solutions to the equation are $$x = \frac{1}{2}$$ and $$x = -2$$.