Quadratic Equation Solver
Enter a, b, c and get instant roots, the discriminant, the vertex, and a graph — with every step of the quadratic formula shown.
About this tool
What this tool does
Give it the three coefficients of a quadratic equation — ax² + bx + c = 0 — and it solves for x using the quadratic formula, showing the discriminant, both roots (real or complex), the vertex, the axis of symmetry, and a graph of the parabola. You can type the coefficients directly or paste something like "2x^2 - 5x + 3 = 0" and let it parse the numbers out for you.
How to use it
Enter a, b, and c and the solution updates instantly — no submit button needed, though there's one if you prefer it. The step-by-step panel shows the actual substitution into the quadratic formula, not just the final answer, so you can check your own work line by line. The graph plots the parabola with the vertex and any real roots marked, which makes it obvious at a glance whether the discriminant sign matches what you'd expect from the curve's shape.
Common use cases
- Checking algebra homework or exam prep for a step you can't get to match
- Quickly finding where a projectile-motion or optimization equation crosses zero
- Verifying factoring by hand against the formula-derived roots
- Visualizing how changing a, b, or c shifts and reshapes the parabola
Frequently asked questions
What does the discriminant tell you?
Its sign tells you the nature of the roots before you even finish solving. Positive means two distinct real roots, zero means one repeated real root (the parabola just touches the x-axis at its vertex), and negative means no real roots — the parabola never crosses the x-axis, and the roots are complex.
What if a is 0?
Then it's not a quadratic equation anymore — it's linear (bx + c = 0), and the quadratic formula divides by zero. The tool will flag this rather than give you a meaningless answer.
What are complex roots, in plain terms?
They're the solutions you get when the discriminant is negative — the square root of a negative number isn't a real number, so the two roots are written as a real part plus or minus an imaginary part (using i, where i² = -1). Graphically, this is exactly the case where the parabola stays entirely above or below the x-axis.
How is the vertex calculated?
The x-coordinate is -b/2a (also the axis of symmetry), and plugging that back into the original equation gives the y-coordinate. It's the minimum point if a is positive, or the maximum if a is negative.
Can I solve an equation that isn't already set to zero?
Rearrange it first — move everything to one side so it equals zero, matching the ax² + bx + c = 0 form. The equation parser expects that standard form when you paste text in.