Quadratic Formula Solver

What this solver is for

A quadratic equation has the form ax² + bx + c = 0, where a is not zero. This tool takes your three numbers — a, b, and c — and works through the quadratic formula for you: x = (−b ± √(b² − 4ac)) / 2a. Instead of just handing back an answer, it shows the discriminant, both roots, and the vertex of the parabola, so you can check the reasoning rather than trust a black box.

When you'd use it

Algebra and precalculus homework is the obvious case, but quadratics show up well beyond the classroom. Physics uses them for projectile height and time-of-flight. Engineers solve them for areas and load. If you're finding where a curve crosses the x-axis, calculating a break-even point, or checking work you did by hand, this saves the arithmetic and catches sign mistakes.

How the discriminant tells the story

The part under the square root, b² − 4ac, is the discriminant, and it decides everything before you finish. If it's positive, you get two distinct real roots. If it's exactly zero, the two roots merge into one repeated root and the parabola just touches the x-axis. If it's negative, there are no real roots — the answers are a complex conjugate pair, which this tool reports as p ± qi. For example, x² − 5x + 6 = 0 has a discriminant of 1 and roots 2 and 3, while x² + 4 = 0 has a discriminant of −16 and roots ±2i.

How to use it

1. Rearrange your equation so one side equals zero, then read off a (the x² coefficient), b (the x coefficient), and c (the constant).
2. Type those three values into the a, b, and c fields. Use a negative sign for negative terms, and enter decimals or fractions like 0.5 directly.
3. Read the results: the discriminant, the real or complex roots, and the vertex (h, k) where the parabola turns.
4. Change any coefficient to instantly re-solve — handy for exploring how shifting c moves the roots.

FAQ

What if I leave a as zero?

Then it isn't quadratic — it's the linear equation bx + c = 0, and the x² term disappears. This solver needs a to be non-zero to apply the quadratic formula.

Why does it show the vertex too?

The vertex is the highest or lowest point of the parabola, found at x = −b/2a. It's useful for graphing and for finding maximum or minimum values, which the roots alone don't tell you.

Can it handle answers that aren't whole numbers?

Yes. Irrational roots (with a square root) and complex roots are both supported, and results are shown in exact or decimal form rather than being rounded away.

Is it free, and does my equation get sent anywhere?

It's completely free, supported by ads. Every calculation runs inside your browser, so nothing you type is uploaded or stored.