Vertex Form Calculator

Use vertex form a(x - h)^2 + k whenever you care about the shape of the parabola: the vertex (h, k), the axis of symmetry (x = h), the direction it opens (sign of a), and the maximum or minimum value (k itself). Graphing, transformations, and optimization problems ("what input gives the largest output?") all read straight off vertex form. Standard form ax^2 + bx + c is the right pick when you want to read off the y-intercept (it is just c) or when you plan to feed the coefficients into the quadratic formula to find the roots. The two forms describe the same parabola, so pick the one that matches the question.

Start with ax^2 + bx + c, factor a from the first two terms, then add and subtract (b/(2a))^2 inside the parentheses. This produces a(x - h)^2 + k with h = -b/(2a) and k = c - b^2/(4a). The calculator applies those formulas with JavaScript floating-point arithmetic and rounds the displayed values.

The discriminant b^2 - 4ac decides how the parabola meets the x-axis. If it is positive, the parabola crosses the x-axis at two distinct points, so the equation has two real roots. If it is zero, the parabola just touches the x-axis at its vertex, giving one repeated real root (the vertex sits on the x-axis). If it is negative, the parabola sits entirely above or below the x-axis and the two roots are complex conjugates of the form p +/- q*i. You can read "how many real solutions does this quadratic have?" off the sign of one number.

Both forms divide by a (vertex form puts a out front; standard-to-vertex uses h = -b/(2a) and k = c - b^2/(4a)). If a = 0 the x^2 term disappears entirely and what is left, bx + c, is a linear equation, not a quadratic. A line has no vertex, no axis of symmetry, and at most one root, so none of the outputs the calculator returns would make sense. The calculator returns no result for a = 0 and you can solve the leftover linear case directly with x = -c/b.

The quadratic formula contains a square root of the discriminant. When the discriminant is negative, the square root of a negative number is not a real number; it is defined using the imaginary unit i, where i*i = -1. The two roots then take the form p +/- q*i where p = -b/(2a) is the real part (which is also the axis of symmetry) and q = sqrt(-(b^2 - 4ac)) / (2|a|) is the imaginary part. Geometrically, the parabola never crosses the x-axis: it lives entirely above it (if a > 0 and the vertex is above) or entirely below it (if a < 0 and the vertex is below), so no real x can satisfy ax^2 + bx + c = 0.