Quadratic Equation Calculator
Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, discriminant, vertex, axis of symmetry, factored form and a graph of the parabola.
Quadratic Equation Calculator: with the default inputs, solutions is x = 1 or x = 2.
Can't be 0 — that would make the equation linear.
- Type of roots
- Two distinct real roots
- Discriminant (b² − 4ac)
- 1
- x₁
- 1Blank when the roots are complex.
- x₂
- 2
- Factored form
- (x − 1)(x − 2)
- Vertex
- (1.5, −0.25)
- Axis of symmetry
- x = 1.5
Assumptions
- Roots are computed with the numerically stable form of the formula to avoid cancellation error.
- Factored form with whole coefficients is shown when both roots are rational with denominators up to 1,000.
How this is worked out
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a Discriminant D = b² − 4ac: D > 0 two real roots, D = 0 one repeated root, D < 0 two complex roots Vertex: x = −b ÷ 2a, y = c − b² ÷ 4a Sum of roots = −b/a, product of roots = c/a
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- a (coefficient of x²)
- Can't be 0 — that would make the equation linear.defaults to 1
- b (coefficient of x)
- A number.defaults to -3
- c (constant)
- A number.defaults to 2
What you get back
- Solutionsmain answer
- Type of roots
- Discriminant (b² − 4ac)
- x₁
- Blank when the roots are complex.
- x₂
- Factored form
- Vertex
- Axis of symmetry
What this assumes
- Roots are computed with the numerically stable form of the formula to avoid cancellation error.
- Factored form with whole coefficients is shown when both roots are rational with denominators up to 1,000.
About this calculator
A quadratic equation is any equation that can be written as ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and solving the equation means finding where that parabola crosses the x-axis. The quadratic formula does this for every quadratic, no factoring luck required, and this calculator shows each stage of it.
How to use it
Enter a, b and c. Move everything to one side first: 2x² = 5x − 3 becomes 2x² − 5x + 3 = 0, so a = 2, b = −5, c = 3. The results give the roots, the discriminant, the factored form when one exists, the vertex and the axis of symmetry, plus a graph so you can see why the roots are where they are.
Where the formula comes from
Completing the square on ax² + bx + c = 0 gives (x + b/2a)² = (b² − 4ac)/4a². Taking the square root of both sides and solving for x produces the formula. The piece under the root, b² − 4ac, is the discriminant, and it tells the whole story before you finish:
- D > 0: two different real roots — the parabola crosses the axis twice.
- D = 0: one repeated root — the vertex sits exactly on the axis.
- D < 0: no real roots — the parabola floats entirely above or below the axis. The roots are complex conjugates like 1 ± 2i, which this calculator reports rather than saying "no solution".
Vertex and symmetry
The parabola is symmetric about the vertical line x = −b/2a, which is always the midpoint of the two roots. The vertex is the highest or lowest point (lowest when a > 0). For anything you're maximizing or minimizing — profit, height of a thrown ball — the vertex is the answer.
Factored form
When the roots are rational the calculator rebuilds the factored form with whole-number coefficients, e.g. 2x² + 5x + 3 = (2x + 3)(x + 1). If the roots are irrational it shows a(x − r₁)(x − r₂) with decimals; if they're complex there's no real factorization.
Common mistakes
- Sign errors on b: for x² − 3x + 2, b is −3, so −b is +3.
- Dividing only part of the numerator by 2a. The whole −b ± √D is divided.
- Forgetting the ± and reporting only one root.
- Entering a = 0. That's a linear equation; the calculator tells you the linear solution instead.
Frequently asked questions
▸What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both solutions of ax² + bx + c = 0 for any coefficients with a ≠ 0.
▸What does the discriminant tell me?
b² − 4ac decides how many real solutions there are: positive means two, zero means one repeated root, negative means none (two complex roots).
▸What if a is zero?
Then there is no x² term and the equation is linear: bx + c = 0, so x = −c/b. The calculator shows that answer in its error message.
▸How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a; plug it back in for y (or use y = c − b² ÷ 4a). For x² − 3x + 2 the vertex is (1.5, −0.25).
▸Can a quadratic have complex solutions?
Yes, whenever the discriminant is negative. The roots come as a conjugate pair p ± qi, and the calculator writes them that way.
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