The quadratic equation is \( 2x^2 - 4x - 6 = 0 \), where \( a = 2 \), \( b = -4 \), and \( c = -6 \).
["# Solving the Quadratic Equation: ( 2x^2 - 4x - 6 = 0 )", "The quadratic equation plays a fundamental role in algebra, providing key insights into parabolic relationships, real-world problems, and graphical behavior. One commonly encountered quadratic equation is:", "[\n2x^2 - 4x - 6 = 0\n]", "Here, the coefficients are clearly defined as:\n- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "This equation fits perfectly into the standard quadratic form ( ax^2 + bx + c = 0 ), enabling efficient solving through factoring, completing the square, or using the quadratic formula.", "---", "## Why Solve Quadratic Equations?", "Quadratic equations help model scenarios involving area, projectile motion, optimization, and more. Understanding how to solve equations like ( 2x^2 - 4x - 6 = 0 ) equips students, engineers, and scientists with critical problem-solving skills.", "---", "## Step-by-Step Solution Using the Quadratic Formula", "The quadratic formula is the most reliable method for solving any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 1: Identify coefficients\nWe have:\n- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "### Step 2: Calculate the discriminant\nThe discriminant ( D = b^2 - 4ac ):", "[\nD = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\n]", "Since ( D > 0 ), we expect two distinct real solutions.", "### Step 3: Plug values into the quadratic formula", "[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4}\n]", "### Step 4: Calculate the two solutions\n[\nx_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]\n[\nx_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "---", "## Final Result", "The solutions to the equation ( 2x^2 - 4x - 6 = 0 ) are:", "[\nx = 3\quad \ ext{and} \quad x = -1\n]", "These points represent where the quadratic function ( y = 2x^2 - 4x - 6 ) crosses the ( x )-axis—also known as the roots or zeros.", "---", "## Tips for Quick Factoring (When Possible)", "Although not necessary here, if you try factoring:", "[\n2x^2 - 4x - 6 = 2(x^2 - 2x - 3) = 2(x - 3)(x + 1)\n]", "Setting each factor to zero confirms:\n( x - 3 = 0 \Rightarrow x = 3 )\n( x + 1 = 0 \Rightarrow x = -1 )", "---", "## Visualizing the Quadratic Function", "The graph of ( y = 2x^2 - 4x - 6 ) is a parabola opening upward (since ( a = 2 > 0 )), intersecting the ( x )-axis at ( x = -1 ) and ( x = 3 ). This location helps interpret the real-world context of applications like profit models and physics problems.", "---", "## Conclusion", "Mastering quadratic equations through precise coefficient identification, discriminant evaluation, and solution application strengthens algebraic reasoning. The equation ( 2x^2 - 4x - 6 = 0 ) yields concise, meaningful solutions that reflect both mathematical rigor and practical relevance—essential for students and professionals alike.", "---", "Keywords: quadratic equation, solve (2x^2 - 4x - 6 = 0), quadratic formula, discriminant, roots, real solutions, algebra tutorial, math practice, quadratic roots."]
