Vertex x-Coordinate of a Parabola

Every parabola y = ax² + bx + c is symmetric about a single vertical line, and that line runs through its turning point at x = −b/(2a). The expression is just the quadratic formula with the square root deleted: the two roots sit at −b/(2a) plus and minus √Δ/(2a), so their midpoint — the vertex — is the shared first term. The equation of the axis of symmetry is literally x = h. Worked example: y = 2x² − 8x + 5 turns at h = 8/4 = 2, and its roots 2 ± √(6)/2 straddle that value symmetrically.

This is the workhorse of every optimisation question in a first algebra course, because a parabola's maximum or minimum can only occur at the vertex: fence 40 m of wire into the largest rectangle, and the area A = x(20 − x) = −x² + 20x peaks at h = −20/(2 × −1) = 10 m, a square. Descartes' La Géométrie of 1637 is what made this arithmetic possible at all — before coordinates, a parabola was a slice through a cone and its axis was found by construction, not by dividing two numbers. The trap is dropping the minus sign or forgetting the 2 in the denominator; if a is negative, remember that h can still be positive because two negatives cancel.