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.
Equation: x² − 3x + 2 = 0 Discriminant: 1 Nature: Two distinct real roots Roots: 2, 1 Vertex: (1.5, -0.25) Axis of symmetry: x = 1.5 Sum of roots: 3 Product of roots: 2 y-intercept: (0, 2) Parabola: opens upward (a > 0), lowest at the vertex; crosses the x-axis at the two roots
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
- Factored form
- (x − 2)(x − 1)
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. It ends with the y-intercept and which way the parabola opens — and whether it crosses the x-axis, touches it at the vertex or never meets it — and "Reset" puts back the equation the page opens on. 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 — rather than silently turning your quadratic into a linear equation, the page names the box and waits.
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. 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. Leaving the box empty is different again: the page tells you which box is empty instead of guessing that you meant zero.
Does it give exact roots and the axis of symmetry?
Yes. When the roots are irrational it writes them exactly, with square roots, beside the decimals. The axis of symmetry is given and drawn on the graph, and the working ends in the factored form.
Does it say which way the parabola opens, or where it meets the y-axis?
Yes. The answer ends with the y-intercept, (0, c), and a line saying whether the parabola opens upward or downward, whether the vertex is its lowest or highest point, and whether it crosses the x-axis at two roots, touches it at the vertex or never meets it.
Can I save the graph for homework or slides?
Yes. Under the parabola, “Download graph (SVG)” saves it as a vector file and “Download graph (PNG)” as an image drawn at double resolution for sharpness. Both have a white background and fixed colours whatever the page's theme, with the equation written above the curve, and show the same roots, vertex and axis as the graph on the page.