Use the fact that the sum of the roots of \( ax^2 + bx + c = 0 \) is \(-\frac{b}{a}\). Here, \( a = 2 \), \( b = -4 \):

["Understanding Quadratic Equations: Using the Sum of Roots Formula", "When studying quadratic equations, one of the most powerful tools students and math enthusiasts can use is the relationship between a quadratic’s coefficients and the sum of its roots. For any quadratic equation in the standard form:", "[\nax^2 + bx + c = 0\n]", "the sum of the roots (also known as the roots of the equation) is given by the formula:", "[\n\ ext{Sum of roots} = -\frac{b}{a}\n]", "This elegant relationship not only simplifies solving problems but also offers deep insight into the structure of quadratic functions. Let’s explore how this formula applies using specific values — with ( a = 2 ) and ( b = -4 ).", "### Why Knowing the Sum of Roots Matters", "While finding the exact roots of a quadratic equation is often the goal, sometimes knowing just the sum of the roots helps in:", "- Verifying solutions quickly without solving the entire equation\n- Checking work when solving quadratic equations\n- Understanding symmetry and behavior of parabolas defined by ( ax^2 + bx + c = 0 )", "The sum formula rests on Vieta’s formulas, which connect coefficients to sum and product of roots. The sum of the roots tells us crucial information about where the graph crosses the x-axis.", "---", "### Applying the Formula with ( a = 2 ), ( b = -4 )", "Given:\n- ( a = 2 )\n- ( b = -4 )", "Plug these into the sum of roots formula:", "[\n\ ext{Sum of roots} = -\frac{b}{a} = -\frac{-4}{2} = \frac{4}{2} = 2\n]", "So, if the quadratic equation has coefficients ( a = 2 ) and ( b = -4 ), the sum of its roots is 2.", "---", "### Real-World Example and Application", "Imagine a projectile launched vertically following the path ( y = 2x^2 - 4x + c ), where ( x ) is time in seconds and ( y ) is height in meters.", "From above, the combined value of the roots (in time) is 2 seconds. This means that if the projectile starts at ground level and hits the ground at two distinct times, those times add up to 2 seconds.", "While we don’t know ( c ) without restricting the roots, knowing the sum helps set bounds or set up equations if additional conditions are given — making this formula indispensable in modeling quadratic behaviors.", "---", "### Why This Formula Is a Math Champion", "The sum of roots formula (( -\frac{b}{a} )) bridges algebra and geometry:", "- It simplifies calculations compared to solving quadratic equations via the quadratic formula.\n- It quickly reveals properties like symmetry about the vertex's x-coordinate.\n- It supports problem-solving in applications such as physics, economics, and engineering.", "---", "### Quick Summary", "| Given | Value |\n|-------|-------|\n| ( a ) | 2 |\n| ( b ) | -4 |\n| Sum of roots = | ( -\frac{b}{a} = 2 ) |", "---", "### Final Thoughts", "Mastering Vieta’s formulas — especially the sum of roots — empowers learners to work more efficiently with quadratic equations. Whether you're a student tackling algebra homework or a professional modeling real-world phenomena, the fact that the sum of roots equals (-\frac{b}{a}) is a key shortcut that saves time and deepens understanding.", "Next time you encounter a quadratic equation, pause — it often reveals the sum of its roots instantly!", "---", "Keywords: quadratic equation, sum of roots formula, Vieta’s formulas, ( ax^2 + bx + c = 0 ), ( a = 2 ), ( b = -4 ), algebra shortcut, quadratic roots, parabola, mathematical application"]