A quadratic equation is given by \( 2x^2 - 4x - 6 = 0 \). Solve for \( x \) using the quadratic formula.
["# Solving Quadratic Equations: How to Solve ( 2x^2 - 4x - 6 = 0 ) Using the Quadratic Formula", "Quadratic equations are a fundamental part of algebra, and learning how to solve them is essential for students and math enthusiasts alike. In this article, we’ll walk through solving the quadratic equation:", "[ 2x^2 - 4x - 6 = 0 ]", "using the standard quadratic formula. We’ll explain each step clearly, helping you understand not just what the solution is, but how we arrive at it.", "## Understanding the Quadratic Formula", "The quadratic formula solves equations of the form:", "[ ax^2 + bx + c = 0 ]", "Where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). The formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides two solutions: one using the “plus” symbol (( + )) and one using the “minus” symbol (( - )). The term under the square root, ( b^2 - 4ac ), is called the discriminant—it determines the nature of the roots (real and distinct, real and repeated, or complex).", "## Step 1: Identify the coefficients ( 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 )", "## Step 2: Compute the discriminant", "Calculate ( b^2 - 4ac ) to determine the type of solutions:", "[\nb^2 = (-4)^2 = 16\n]\n[\n4ac = 4 \ imes 2 \ imes (-6) = -48\n]\n[\nb^2 - 4ac = 16 - (-48) = 16 + 48 = 64\n]", "Since ( 64 > 0 ), there are two distinct real solutions.", "## Step 3: Plug values into the quadratic formula", "[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 2} = \frac{4 \pm 8}{4}\n]", "Now compute both solutions:", "- First solution (( +\sqrt{64} )):\n[\nx = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]", "- Second solution (( -\sqrt{64} )):\n[\nx = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "## Step 4: Final solutions", "The solutions to the equation ( 2x^2 - 4x - 6 = 0 ) are:", "[\nx = 3 \quad \ ext{and} \quad x = -1\n]", "## Summary", "Solving quadratic equations step by step using the quadratic formula is a powerful technique. For the equation ( 2x^2 - 4x - 6 = 0 ), we found:", "- ( a = 2 ), ( b = -4 ), ( c = -6 )\n- Discriminant = 64 (positive, two real solutions)\n- Solutions: ( x = 3 ) and ( x = -1 )", "Understanding this process strengthens your ability to tackle similar problems and deepens your grasp of algebraic methods. Keep practicing—mastery comes with repetition and clear steps!", "If you found this guide helpful, share it with fellow learners and explore more on solving quadratics using factoring, completing the square, and graphing techniques.", "---", "Keywords: quadratic equation, solve (2x^2 - 4x - 6 = 0), quadratic formula, mathematical methods, algebra tutorial, real solutions to quadratics, step-by-step solution, discriminant, equation solving, math help, high school math."]
