Maximum à \( x = -rac{b}{2a} = -rac{120}{2(-2)} = 30 \).

["Maximum of a Quadratic Function: Understanding the Vertex Formula – Solving for ( x = 30 )", "When studying quadratic functions, one of the most important concepts is identifying the vertex — the point where the graph reaches its maximum or minimum value. For a quadratic equation in standard form, ( f(x) = ax^2 + bx + c ), the x-coordinate of the vertex can be efficiently found using Maximum à x = -b/(2a). This formula is not only central to algebraic problem-solving but also pivotal in fields like physics, economics, and engineering where optimization is key.", "### The Quadratic Formula in Focus\nA quadratic equation is written as:\n[\nf(x) = ax^2 + bx + c\n]\nThe coefficient ( a ) determines the parabola’s direction (upward if ( a > 0 ), downward if ( a < 0 )), while ( b ) and ( c ) influence the shape and vertical position. Since the vertex represents the peak (maximum) or trough (minimum), knowing the x-value of the vertex allows us to determine the function’s turning point.", "### Deriving the Maximum x-Value: ( x = -\frac{b}{2a} )", "To find the vertex’s x-coordinate, observe the symmetry of the parabola: the vertex lies exactly halfway between the two roots (zeros) of the equation. This midpoint formula naturally leads to:\n[\nx = -\frac{b}{2a}\n]\nFor example, consider the quadratic:\n[\nf(x) = -2x^2 + 120x\n]\nHere, ( a = -2 ), ( b = 120 ). Plugging into the formula:\n[\nx = -\frac{b}{2a} = -\frac{120}{2(-2)} = -\frac{120}{-4} = 30\n]\nThus, the maximum value occurs at ( x = 30 ). This result reflects the function’s peak due to the downward parabola (since ( a < 0 )).", "### Why ( x = 30 ) Represents Maximum Output\nIn this case, because ( a = -2 ) is negative, the parabola opens downward, confirming that the vertex at ( x = 30 ) is indeed a maximum. At ( x = 30 ), the function reaches its highest point — crucial when modeling real-world systems such as profit maximization, projectile motion, or minimizing costs.", "### Applications Beyond Algebra\nUnderstanding this formula’s derivation and significance extends well beyond homework problems. Engineers use the vertex to optimize structural designs, economists apply it to maximize revenue curves, and scientists model trajectories. Mastery of the formula ( x = -\frac{b}{2a} ) equips learners to solve complex real-life optimization challenges with confidence.", "### Conclusion\nThe vertex’s x-coordinate, calculated as ( x = -\frac{b}{2a} ), is foundational in quadratic analysis. For the equation ( f(x) = -2x^2 + 120x ), this yields ( x = 30 ), the point of maximum value. Grasping this concept simplifies advanced problem-solving and opens doors to practical applications across disciplines.", "Optimize your understanding — anytime you encounter a quadratic, remember: ( x = -\frac{b}{2a} ) points to the peak, whether in math class or everyday decisions.", "---", "Keywords: Quadratic vertex, Maximum of quadratic function, x = -b/(2a), Algebra 101, Parabola optimization, Quadratic formulas explained."]