Solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula.

["# Solve the Quadratic Equation ( 2x^2 - 4x - 6 = 0 ) Using the Quadratic Formula", "Quadratic equations are fundamental in algebra and appear frequently in various scientific and engineering applications. One common method to find the solutions of a quadratic equation is the quadratic formula, which provides an efficient way to solve equations of the form ( ax^2 + bx + c = 0 ). In this article, we’ll learn how to solve the equation ( 2x^2 - 4x - 6 = 0 ) step by step using the quadratic formula.", "## What is the Quadratic Formula?", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Where ( a ), ( b ), and ( c ) are the coefficients from the standard quadratic equation:", "[\nax^2 + bx + c = 0\n]", "This formula calculates the two (possibly repeated) real roots of the quadratic equation, depending on the discriminant ( b^2 - 4ac ).", "## Step-by-Step: Solving ( 2x^2 - 4x - 6 = 0 )", "### 1. Identify ( a ), ( b ), and ( c )", "From the equation ( 2x^2 - 4x - 6 = 0 ), compare with ( ax^2 + bx + c = 0 ):", "- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "### 2. Compute the Discriminant", "The discriminant ( D ) determines the nature of the roots:", "[\nD = b^2 - 4ac\n]", "Substitute the values:", "[\nD = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\n]", "Since ( D = 64 > 0 ), there are two distinct real roots.", "### 3. Apply the Quadratic Formula", "[\nx = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4}\n]", "### 4. Calculate the Two Solutions", "- First root (( + ) sign):", "[\nx_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]", "- Second root (( - ) sign):", "[\nx_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "## Final Answer", "The solutions to the quadratic equation ( 2x^2 - 4x - 6 = 0 ) are:", "[\n\boxed{x = 3 \quad} \ ext{and} \quad x = -1\n]", "## Why Use the Quadratic Formula?", "- It provides a direct, reliable method regardless of how complicated the coefficients are.\n- It clearly shows whether solutions are real, complex, or repeated via the discriminant.\n- It’s essential for solving equations where factoring isn’t straightforward.", "## Conclusion", "Using the quadratic formula, we efficiently solve ( 2x^2 - 4x - 6 = 0 ) by identifying ( a = 2 ), ( b = -4 ), and ( c = -6 ), computing the discriminant, and plugging values into the formula. The results are ( x = 3 ) and ( x = -1 ). This structured approach ensures accurate and quick solutions every time.", "For students, math learners, or STEM professionals, mastering this method builds a strong foundation for tackling advanced algebra and real-world problems involving quadratic relationships.", "---", "Keywords: solve quadratic equation, quadratic formula, solve 2x² - 4x - 6 = 0, quadratic roots, algebra tutorial, discriminant analysis, solve quadratic using formula"]