Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -4 \), \( c = -6 \).

["# Solving Quadratic Equations with Ease: Using the Quadratic Formula", "Understanding and solving quadratic equations is a foundational skill in algebra, essential for students, educators, and anyone tackling real-world math problems. One of the most reliable tools for solving any quadratic equation is the quadratic formula. In this article, we’ll walk through solving the specific quadratic equation using the standard formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "with coefficients ( a = 2 ), ( b = -4 ), and ( c = -6 ). Whether you're a student preparing for exams or a professional needing a quick reference, mastering this method empowers you to solve quadratic equations efficiently.", "## Why the Quadratic Formula Matters", "Quadratic equations take the standard form:\n[\nax^2 + bx + c = 0\n]\nThey model various phenomena such as projectile motion, profit optimization, and geometric problems. While some equations can be factored, many cannot — making the quadratic formula indispensable. It provides exact solutions regardless of whether roots are rational or irrational, real or complex.", "## Plugging in the Values", "Given:\n[\na = 2,\quad b = -4,\quad c = -6\n]", "Substitute these values into the quadratic formula:\n[\nx = \frac{-({-4}) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)}\n]", "Now simplify step by step:", "1. Simplify the numerator term ( -b ):\n[\n -(-4) = +4\n]", "2. Calculate the discriminant (( \Delta = b^2 - 4ac )):\n[\n(-4)^2 = 16\n]\n[\n4ac = 4(2)(-6) = -48\n]\n[\n\Delta = 16 - (-48) = 16 + 48 = 64\n]", "3. Take the square root of the discriminant:\n[\n\sqrt{64} = 8\n]", "4. Substitute into the formula:\n[\nx = \frac{4 \pm 8}{4}\n]", "## Solving for Both Roots", "Now compute the two possible solutions using ( \pm ):", "- First root (( + )):\n[\nx = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]", "- Second root (( - )):\n[\nx = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "## Final Answer", "The solutions to the quadratic equation with ( a = 2, b = -4, c = -6 ) are:\n[\n\boxed{x = 3} \quad \ ext{and} \quad \boxed{x = -1}\n]", "## Conclusion", "The quadratic formula is a powerful, universal tool for solving quadratic equations. By following the steps—substituting coefficients, computing the discriminant, and evaluating both signs—you can efficiently find real or complex roots. Mastering this process streamlines learning and problem-solving in algebra and beyond.", "Whether you're studying for a test, solving engineering challenges, or analyzing data trends, using ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) ensures accuracy and builds confidence in tackling quadratic equations.", "---\nKeywords: quadratic formula, solve quadratic equations, vertex, roots of quadratic, discriminant, algebra tutorial, projectile motion model, real-world applications, mathematical solutions."]