Quadratic Calculator
Enter a, b and c to solve ax² + bx + c = 0: roots, discriminant, vertex, factored form and a graph, with the quadratic formula worked through step by step.
- Roots and vertex
- With steps
- Graph
- Private
Enter the coefficients of a·x² + b·x + c = 0. Negative numbers and decimals are fine; a must not be 0.
Step by step
How to solve a quadratic equation
Three coefficients are all you need.
- 1
Bring the equation into standard form
Rearrange it to ax² + bx + c = 0, so that one side is zero.
- 2
Enter a, b and c
Type the three coefficients, including the minus signs. Decimals are allowed.
- 3
Read the solutions
You get the roots, the discriminant, the vertex, other forms of the equation, the steps and the graph.
Everything about the parabola in one view
For algebra homework and for checking your own working.
All kinds of roots
Two real roots, one repeated root or two complex roots — each case is recognised and explained.
Exact form
Roots such as 1 ± √2 are shown exactly as well as in decimals.
Vertex and symmetry
The turning point of the parabola, its axis of symmetry and whether it opens up or down.
Other forms
Factored form a(x − x₁)(x − x₂) and vertex form a(x − h)² + k.
Graph
The parabola with its roots and vertex marked.
Link to your calculation
Copy a link that reopens the calculator with your numbers filled in — handy for sharing or saving a scenario.
Made for phones
Large fields, the right keyboard for numbers and a layout that fits small screens.
Private by design
Everything is calculated in your browser. Your numbers are not sent to a server.
The quadratic formula explained
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions are given by the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x + 2 = 0 we have a = 1, b = −3 and c = 2, so x = (3 ± √(9 − 8)) ÷ 2 = (3 ± 1) ÷ 2, which gives x = 1 and x = 2.
The expression under the root, D = b² − 4ac, is the discriminant. It tells you the kind of solutions before you calculate them: if D is positive there are two different real roots, if D is zero there is one repeated root, and if D is negative the roots are two complex conjugates and the parabola does not cross the x-axis.
The graph of y = ax² + bx + c is a parabola. Its vertex lies at x = −b ÷ 2a, exactly halfway between the roots, and it opens upward when a is positive and downward when a is negative. By Vieta’s formulas the roots add up to −b ÷ a and multiply to c ÷ a — a quick way to check your result.
What the discriminant tells you
| Discriminant | Solutions | Example |
|---|---|---|
| D > 0, perfect square | Two rational roots | x² − 3x + 2 = 0 → 1 and 2 |
| D > 0, not a square | Two irrational roots | x² − 2x − 1 = 0 → 1 ± √2 |
| D = 0 | One repeated root | x² + 2x + 1 = 0 → −1 |
| D < 0 | Two complex roots | x² + 2x + 2 = 0 → −1 ± i |
Tips
- Linear, cubic or other equations and systems are handled by the Equation Calculator.
- Need to evaluate √ or powers along the way? Use the Scientific Calculator.
- Coefficients given as fractions? Simplify them first with the Fraction Calculator.
- Before using the formula, check whether the equation factors easily: x² − 5x + 6 = (x − 2)(x − 3) can be solved at a glance.
Frequently asked questions
Can’t find your answer? Contact us — we reply quickly.
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives the solutions of any equation of the form ax² + bx + c = 0 with a ≠ 0.
What is the discriminant?
The value D = b² − 4ac. If D > 0 there are two real solutions, if D = 0 there is one, and if D < 0 there are two complex solutions.
How do I find the vertex of a parabola?
The x-coordinate is h = −b ÷ 2a; substitute it into the equation to get k. For y = x² − 3x + 2 the vertex is (1.5, −0.25).
What if a is 0?
Then the x² term disappears and the equation is linear: bx + c = 0, with the single solution x = −c ÷ b. Use the equation calculator for that.
What are complex roots?
When D is negative, the square root of a negative number appears. The roots are then written with the imaginary unit i, where i² = −1, for example −1 + i and −1 − i.
Are my numbers stored or sent anywhere?
No. The calculation runs in your browser. Your last inputs are remembered on your own device only, so the form is still filled in when you come back; use Reset to clear them.
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