Quadratic Equation Calculator

The vertex form calculator converts a quadratic between two equivalent representations (standard form ax^2 + bx + c and vertex form a(x - h)^2 + k) so you can read the vertex straight off the formula. This calculator is an equation solver: it sets ax^2 + bx + c equal to 0 and finds the values of x that make that true, then reports the discriminant interpretation, the roots themselves (real or complex), the vertex, axis of symmetry, factored form, parabola direction, and y-intercept. Use vertex form when you want to graph or describe the parabola, and use this one when you want to know which x values are zeros of the polynomial.

The discriminant is D = b^2 - 4ac and it decides the kind of solution before you compute it. If D is positive, the parabola crosses the x-axis twice, so there are two distinct real roots. If D is zero, the parabola touches the x-axis exactly once at its vertex, giving one repeated real root. If D is negative, the parabola never meets the x-axis, so the two roots are complex conjugates of the form p +/- q*i. The calculator surfaces D and a plain-language interpretation alongside the roots so you can sanity-check the answer at a glance.

A quadratic with two real roots r1 and r2 factors cleanly into a*(x - r1)*(x - r2), and the calculator prints that product so you can verify the equation by multiplying out. When the discriminant is negative there are no real roots, so no product of linear factors with real coefficients reproduces the polynomial. The technical term is that the quadratic is irreducible over the real numbers. It still factors over the complex numbers as a*(x - (p + q*i))*(x - (p - q*i)), but that form is rarely useful in homework or engineering contexts, so the calculator reports the plain label 'irreducible over reals' instead.

All numeric outputs (roots, discriminant, vertex, axis of symmetry, y-intercept) are rounded to four decimal places for display. The internal math uses full IEEE 754 double precision, but rounding the display avoids long runs of meaningless digits like 1.9999999999999998 that appear when the true answer is exactly 2 and floating point loses the last bit. The branch decision (D positive, zero, or negative) uses the raw, unrounded discriminant so a hairline negative value cannot incorrectly flip a repeated root into a complex pair just because the display would have rendered it as 0.0000.

Look only at the leading coefficient a. If a is positive, the parabola opens upward and the vertex (h, k) is the minimum point on the curve. If a is negative, the parabola opens downward and the vertex is the maximum. The calculator labels this directly with an Opens: Upward or Downward field so you do not have to remember the rule. Combined with the vertex coordinates and the y-intercept (which is just c), that is enough information to sketch the curve by hand: plot the vertex, plot the y-intercept, mirror the y-intercept across the axis of symmetry, and you have three points that fix the shape.