Quadratic Equation Solver
Solve any quadratic ax² + bx + c = 0 for its roots, discriminant and vertex, see the parabola drawn, and follow the formula step by step. Free, no upload.
Equation: x² − 3x + 2 = 0 Discriminant: 1 Nature: Two distinct real roots Roots: 2, 1 Vertex: (1.5, -0.25) Sum of roots: 3 Product of roots: 2
The curve
Step by step
- Discriminant
- D = b² − 4ac = (−3)² − 4·1·2 = 1
- Its square root
- √D = 1
- The quadratic formula, substituted
- x = (3 ± 1) / 2
- Which gives
- x₁ = 2, x₂ = 1
- Completing the square
- (x − 1.5)² − 0.25
🔒 Solved instantly in your browser — nothing is uploaded.
The answer, and the shape it belongs to
Enter the three coefficients and the tool gives the roots, the discriminant, what kind of roots those are, the vertex, and the sum and product of the roots — updated as you type. Under that it draws the parabola itself, to the scale the numbers describe, with the roots marked where the curve crosses the axis and the vertex marked at its turning point. A sign typed wrong is obvious in the picture long before it is obvious in a list of numbers.
Step by step, with your numbers in it
The working is written out rather than assumed: the discriminant with the coefficients substituted (D = b² − 4ac = (−3)² − 4·1·2 = 1), its square root, the quadratic formula with those values put in, the roots that come out, and the same equation with the square completed. That last line, a(x − h)² + k, is where the vertex comes from and is what the graph is really showing.
When it is not a quadratic
With a of zero there is no x² term and no parabola, so the tool solves the linear equation instead and says that is what it did. With a and b both zero it says whether every x works or none does. An empty coefficient box is not read as a zero — the page names the box and waits, because silently turning your quadratic into a linear equation is a wrong answer delivered without a word.
Frequently asked questions
What does the discriminant tell me?
It is b² − 4ac, and its sign is the whole story: positive means the curve crosses the axis twice and there are two real roots, zero means it touches once and the root is repeated, and negative means it never reaches the axis and the roots are a complex conjugate pair. The page shows the substitution, so the number can be checked rather than trusted.
Can I see the graph?
Yes. The parabola is drawn under the answers, scaled to fit whatever equation you typed, with a dot on each real root and a ring on the vertex. When there are no real roots the curve simply sits clear of the axis, which is what having no real roots looks like.
Does it show the working?
Yes — the discriminant substituted, its square root, the quadratic formula with your numbers in it, the resulting roots, and the equation with the square completed. That is the form the marks are given for, and it is the same arithmetic the answers above come from.
What happens if a is zero?
Then it is not a quadratic. The tool solves bx + c = 0 instead and labels the result a linear equation, rather than pretending there is a parabola. Leaving the box empty is different again: the page tells you which box is empty instead of guessing that you meant zero.
Is anything sent to a server?
No. Every calculation and the drawing itself happen in your browser. Nothing you type is uploaded or stored.