Solve ax² + bx + c = 0 and see the discriminant, roots, vertex, y-intercept, and a parabola graph.
Step-by-step solution
| Discriminant | Roots | Graph meets x-axis |
|---|---|---|
| D > 0 | Two distinct real roots | Two points |
| D = 0 | One real root (double) | One point (tangent) |
| D < 0 | Two complex roots (p ± qi) | No intersection |
What does the discriminant tell me?
The discriminant D = b² − 4ac describes the nature of the roots without solving the equation. If D > 0, the parabola crosses the x-axis at two points, so there are two distinct real roots. If D = 0, the parabola touches the x-axis at one point — a single repeated root. If D < 0, the parabola does not touch the x-axis and the roots are a complex conjugate pair p ± qi.
How do I read complex roots?
When D < 0, this calculator writes the roots as p ± qi, where p = −b/(2a) is the real part and q = √|D|/(2a) is the imaginary part. The two roots are always complex conjugates, meaning they share the same real part and have opposite imaginary parts.
What is the vertex and how is it found?
The vertex is the turning point of the parabola, located at (h, k) where h = −b/(2a) and k = c − b²/(4a), which is the same as (4ac − b²)/(4a). If a > 0 the vertex is the lowest point, and if a < 0 it is the highest point. The vertical line x = h is the axis of symmetry.
What is vertex form?
Any quadratic ax² + bx + c can be rewritten as a(x − h)² + k, where (h, k) is the vertex. This form makes the vertex obvious and is useful for graphing and for solving by completing the square. Convert from standard form using h = −b/(2a) and k = f(h).
What happens if a = 0?
If a = 0, the equation is no longer quadratic — it becomes linear: bx + c = 0, with a single solution x = −c/b (assuming b ≠ 0). This calculator requires a ≠ 0 and will show an error otherwise. For linear equations, use a linear equation solver instead.
Why does my teacher want an exact answer instead of a decimal?
Decimal answers like 1.41421 are approximations. Exact forms such as √2, 1 ± √3, or 3/2 preserve the underlying structure and avoid rounding errors. This calculator shows decimal values to 6 significant digits; for exact surds, simplify √D by hand or use a symbolic solver.
This quadratic formula calculator solves any equation of the form ax² + bx + c = 0. Enter the three coefficients and the tool returns the discriminant D = b² − 4ac, classifies the nature of the roots, and computes both solutions using x = (−b ± √D) / (2a). When D is negative, the calculator writes the complex conjugate pair p ± qi instead of showing an error. It also reports the vertex (h, k) with h = −b/(2a) and k = (4ac − b²)/(4a), the axis of symmetry, the y-intercept, and the direction in which the parabola opens. A step-by-step panel shows the formula substitution line by line, and an SVG plot marks the roots, vertex, and y-intercept so you can sanity-check the result visually. Example: for x² − 5x + 6 = 0 the discriminant is 1, and the roots are x₁ = 3 and x₂ = 2.