Solve Any Quadratic, Instantly
Enter a, b, and c from ax² + bx + c = 0 and get your roots — including complex roots when the discriminant goes negative. It applies the quadratic formula for you, useful both for checking your own work and for equations that just don't factor nicely.
The Formula Itself
For ax² + bx + c = 0, the solution is x = (−b ± √(b² − 4ac)) / 2a. The piece under the square root, b² − 4ac, is the discriminant, and its sign tells you what you're about to get before you finish the calculation.
Reading the Discriminant
- Positive: two distinct real roots — the parabola crosses the x-axis twice.
- Zero: one repeated real root — the parabola just touches the x-axis at its vertex.
- Negative: two complex conjugate roots — no crossing at all.
Using the Solver
- Enter a, b, and c from your equation.
- It calculates the discriminant and runs the quadratic formula automatically.
- Check the roots — real or complex — and graph the equation if you want to see them visually.
A Quick Worked Example
Take x² − 5x + 6 = 0, so a = 1, b = −5, c = 6. Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1. Positive, so two real roots. Plug in: x = (5 ± 1) / 2, giving x = 3 or x = 2. Check it by factoring directly — (x − 2)(x − 3) = 0 — and you land on the same two answers.
Quick Tip: Always Plug Your Root Back In
Once you have a root, substitute it back into the original equation to confirm it actually equals zero. Using the worked example above, x = 3: 3² − 5(3) + 6 = 9 − 15 + 6 = 0. It checks out.
That check takes about ten seconds and catches nearly every arithmetic slip — a sign error, a misread coefficient — before it turns into a wrong answer on a homework set or a quiz.
Frequently Asked Questions (FAQ)
Does it handle complex roots?
Yes — a negative discriminant gives you the two complex conjugate roots in a + bi form.
What if a = 0?
Then it's not quadratic anymore, it's linear (bx + c = 0), and should be solved that way instead.
Does it show me the discriminant?
Yes, right alongside the roots, so you can see why you got real vs. complex.
What if my equation isn't already set to zero?
Rearrange it first. x² + 3x = 10 becomes x² + 3x − 10 = 0 before you plug in the coefficients.
Is it free?
Yes, no account needed.
Decimals and negatives fine for a, b, c?
Yes, any real number works.
Does it graph the parabola too?
Pair it with the 2D Graphing Calculator to see the curve and confirm where it crosses the x-axis.
What does one repeated root mean?
A discriminant of exactly zero — the parabola just touches the x-axis at its vertex instead of crossing it twice.
Can I find the vertex here too?
The solver focuses on roots specifically; the vertex x-coordinate is −b/2a.
Is the quadratic formula the only option?
No — factoring and completing the square both work and are often quicker on clean equations.