91Ó°ÊÓ

First, look at the equation given: $$2x^2 + 16x - 24 = 0$$. Step 1: Divide the equation by the coefficient of $$x^2$$. $$x^2 + 8x - 12 = 0$$ Step 2: Move the constant term to the other side of the equation. $$x^2 + 8x = 12$$ Step 3: Find the value to complete the square: $$\left(\frac{8}{2}\right)^2 = 4^2 = 16$$ Step 4: Add the value to both sides of the equation. $$x^2 + 8x + 16 = 12 + 16$$ $$x^2 + 8x + 16 = 28$$ Step 5: Rewrite the left side as a perfect square. $$(x + 4)^2 = 28$$ Step 6: Solve for x. $$x + 4 = \pm\sqrt{28} \Rightarrow x = -4 \pm \sqrt{28}$$ So, the solutions are: $$x_1 = -4 + \sqrt{28}, \quad x_2 = -4 - \sqrt{28}$$

The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Where a, b, and c are coefficients of the quadratic equation $$ax^2 + bx + c = 0$$. Step 1: Identify the coefficients. In the given equation $$2x^2 + 16x - 24 = 0$$, we have: $$a = 2, \quad b = 16, \quad c = -24$$ Step 2: Calculate the discriminant $$\Delta$$. $$\Delta = b^2 - 4ac = 16^2 - 4 \cdot 2 \cdot (-24) = 256 + 192 = 448$$ Step 3: Substitute the coefficients and the discriminant into the quadratic formula and solve for x. $$x = \frac{-16 \pm \sqrt{448}}{4} = \frac{-16 \pm 8\sqrt{7}}{4}$$ So, the solutions are: $$x_1 = -4 + 2\sqrt{7}, \quad x_2 = -4 - 2\sqrt{7}$$ These are the same solutions found in the completing the square method.