A quadratic equation is given by \(2x^2 - 4x - 6 = 0\). Find the roots using the quadratic formula.

["# Finding the Roots of the Quadratic Equation (2x^2 - 4x - 6 = 0) Using the Quadratic Formula", "When solving quadratic equations in the form (ax^2 + bx + c = 0), the quadratic formula is an essential tool. For the equation (2x^2 - 4x - 6 = 0), identifying coefficients (a), (b), and (c) allows us to efficiently compute the roots.", "### Step 1: Identify coefficients\nFrom the equation (2x^2 - 4x - 6 = 0):\n- (a = 2)\n- (b = -4)\n- (c = -6)", "### Step 2: Recall the quadratic formula\nThe roots of the quadratic formula are given by:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 3: Calculate the discriminant\nThe discriminant (D = b^2 - 4ac) determines the nature of the roots:\n[\nD = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\n]\nSince (D > 0), the equation has two distinct real roots.", "### Step 4: Substitute into the quadratic formula\n[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4}\n]", "### Step 5: Solve for both roots\n- First root:\n[\nx = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]\n- Second root:\n[\nx = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "### Final Answer\nThe roots of the quadratic equation (2x^2 - 4x - 6 = 0) are:\n[\n\boxed{x = 3 \quad \ ext{and} \quad x = -1}\n]", "Understanding how to apply the quadratic formula empowers students to solve any quadratic equation systematically, whether the roots are real, repeated, or complex. This method is foundational in algebra and critical for further studies in mathematics and science."]