1. The problem is to solve the equation $$2x^2 - 4x - 6 = 0$$ for $x$. 2. We use the quadratic formula to solve equations of the form $$ax^2 + bx + c = 0$$, which is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-4$, and $c=-6$. 3. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 4. Since the discriminant is positive, there are two real solutions. 5. Substitute values into the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 6. Calculate each solution: - $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ - $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 7. Therefore, the solutions to the equation are $$x = 3$$ and $$x = -1$$.