Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 instantly. Get the discriminant, roots, step-by-step working and a parabola graph. Fully usable on mobile.
Relationships between the roots.
How to Use the Quadratic Equation Solver
From coefficients to full step by step working in seconds. Fully usable on mobile.
Every calculation runs inside your own browser. No account, no tracking, nothing ever uploaded.
Quadratic Equation Solver, Free and Online
A quadratic equation solver finds the values of x that satisfy ax squared plus bx plus c equals 0, using the discriminant to determine whether the roots are real or complex. This one shows full step by step working, a live parabola graph, and runs entirely inside your browser.
Solving a quadratic equation by hand is a core algebra skill, but checking your own work quickly is just as valuable. This free quadratic equation solver tool does both. Enter the coefficients a, b and c, press solve, and see the discriminant, both roots, and a full step by step derivation using the quadratic formula, all worked out automatically. Pro Mode adds vertex form, factored form, a full completing the square walkthrough, Vieta's formulas relating the roots to the coefficients, detailed parabola properties, and an interactive graph showing the roots, vertex and y-intercept plotted directly on the curve.
This formula solves any quadratic equation of the form ax² + bx + c = 0, provided a is not zero. The expression under the square root, b² − 4ac, is the discriminant, and it alone determines what kind of roots the equation has.
1Simple vs Pro, What Each Mode Includes
| Feature | Simple Mode | Pro Mode (Free) |
|---|---|---|
| Discriminant, roots and step by step working | Included | Included |
| Real and complex root support | Included | Included |
| Vertex form and factored form | Not included | Included |
| Completing the square, full walkthrough | Not included | Included |
| Vieta's formulas (sum and product of roots) | Not included | Included |
| Full parabola properties and interactive graph | Not included | Included |
| Downloadable TXT summary and PNG graph | TXT only | TXT and PNG |
2The Discriminant, Quick Reference
The mistake that trips up the most students is forgetting that a cannot be zero. If a equals zero, the x squared term vanishes and the equation collapses into bx plus c equals 0, a linear equation with only one solution rather than a quadratic with two. The quadratic formula also divides by 2a, so a value of zero there would make the whole expression undefined. This solver checks for that case and lets you know immediately, rather than silently producing a meaningless result.
3Where the Quadratic Formula Actually Came From
Long before anyone wrote ax² + bx + c = 0, people were already solving equations shaped exactly like it. According to MacTutor History of Mathematics at the University of St Andrews, Babylonian scribes were solving what we would now call quadratic equations as early as 1800 BC, using a method that was essentially completing the square. The important nuance MacTutor points out is that the Babylonians had no actual concept of an equation at all, they worked entirely in terms of practical geometric problems, typically involving lengths and areas, and their answers were always positive quantities since a negative length made no physical sense. It would take many more centuries, through Greek geometry and later Hindu mathematics, before the modern algebraic notation and the general formula used on this page finally took shape.
| Question | Quick Answer |
|---|---|
| Who first solved quadratic style problems? | Babylonian scribes, around 1800 BC |
| What method did they use? | An early form of completing the square |
| Did they think of it as an equation? | No, they had no concept of equation at all |
| What kind of answers did they accept? | Only positive quantities, treated as lengths |
Every calculation runs inside your own browser using JavaScript, nothing is uploaded or sent to a server, and no account or signup is required.
4Frequently Asked Questions
Grounded in MacTutor History of Mathematics' own detailed account of the origin of quadratic solving methods, not the oversimplified claim that the Babylonians literally invented the equation. The full account is available on the University of St Andrews' MacTutor archive. For related tools on this site, see the Logarithm Calculator and the full SmallStudyTools.com tool library.
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