standard form of a quadratic

Standard form is the “tidied up” way to write a quadratic, with the terms lined up from highest power to lowest: the squared term first, then the linear term, then the constant. It is the default form you usually start from before factoring, graphing, or applying a formula.

Concretely, standard form is ax^2 + bx + c (or set equal to zero for an equation). Here a is the leading coefficient, b is the linear coefficient, and c is the constant term. Reading off a, b, c at a glance is exactly what lets you plug straight into the quadratic formula or compute the discriminant.

Standard form is great for spotting coefficients and the y-intercept (which is simply c), but it hides the vertex. To reveal the turning point you convert to vertex form by completing the square. The two forms describe the very same parabola — they just highlight different features.