Solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula \( x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

["Solve the Quadratic Equation ( 2x^2 - 4x - 6 = 0 ) Using the Quadratic Formula", "Quadratic equations are essential in algebra and appear frequently in science, engineering, and mathematics. One common method to solve them is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In this article, we’ll solve the equation:", "[\n2x^2 - 4x - 6 = 0\n]", "using the quadratic formula step-by-step.", "---", "### Step 1: Identify coefficients ( a ), ( b ), and ( c )", "The standard form of a quadratic equation is ( ax^2 + bx + c = 0 ). For the given equation:", "- ( a = 2 ) (coefficient of ( x^2 ))\n- ( b = -4 ) (coefficient of ( x ))\n- ( c = -6 ) (constant term)", "---", "### Step 2: Calculate the discriminant", "The discriminant ( D ) determines the nature of the roots and is computed using:", "[\nD = b^2 - 4ac\n]", "Substitute ( a = 2 ), ( b = -4 ), ( c = -6 ):", "[\nD = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\n]", "Since ( D > 0 ), there are two distinct real roots.", "---", "### Step 3: Apply the quadratic formula", "Now substitute ( a ), ( b ), and ( D ) into the quadratic formula:", "[\nx = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4}\n]", "---", "### Step 4: Calculate the two solutions", "Break into two cases using ( \pm ):", "1. ( x = \frac{4 + 8}{4} = \frac{12}{4} = 3 )\n2. ( x = \frac{4 - 8}{4} = \frac{-4}{4} = -1 )", "---", "### Final Answer", "The solutions to the equation ( 2x^2 - 4x - 6 = 0 ) are:", "[\n\boxed{x = 3 \quad \ ext{and} \quad x = -1}\n]", "---", "### Why Use the Quadratic Formula?", "Using the quadratic formula avoids messy factoring and works reliably for any quadratic equation, whether factorable or not. It’s a powerful tool that simplifies solving one of algebra’s most fundamental types of equations.", "If you’re studying quadratic equations, mastering this method will boost your confidence and accuracy in solving more complex problems.", "---", "Keywords: quadratic equation, solve ( 2x^2 - 4x - 6 = 0 ), quadratic formula, discriminant, real roots, algebra tutorial."]