Solve Second-Degree Equations Instantly
This quadratic equation solver handles any equation of the form ax² + bx + c = 0 using the quadratic formula. Enter the three coefficients and it returns the roots the moment you type — together with the discriminant, the parabola's vertex and axis of symmetry, and a numbered step-by-step solution.
ax² + bx + c = 0 where a ≠ 0. Depending on its discriminant, it can have 0, 1, or 2 real solutions.Common Use Cases
Algebra Homework
Graphing Parabolas
Discriminant Study
How to Solve a Quadratic Equation
Enter the Coefficients
Type the values of a, b, c for ax² + bx + c = 0. A blank field is read as 0, and the coefficient a should be non-zero — otherwise the equation is linear, not quadratic.
Check the Live Preview
The equation preview and solution update in real time as you type, confirming the equation matches what you intended.
Review the Properties
View the discriminant (Δ), the roots, the vertex coordinates, and the axis of symmetry, all computed automatically.
Follow the Steps
Open the step-by-step solution to see the discriminant calculated and the quadratic formula applied to reach each root.
The Quadratic Formula
Every quadratic ax² + bx + c = 0 (with a ≠ 0) is solved with the same formula:
The vertex of the parabola sits at (−b/2a, f(−b/2a)), and its axis of symmetry is the vertical line x = −b/2a. Roots that come out clean are shown as exact fractions, with the decimal value in parentheses.
Worked Examples
| Equation | Discriminant Δ | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 25 − 24 = 1 | x = 2, x = 3 |
| x² − 4x + 4 = 0 | 16 − 16 = 0 | x = 2 (double) |
| x² + 1 = 0 | 0 − 4 = −4 | No real roots |
| 2x² − 3x − 2 = 0 | 9 + 16 = 25 | x = 2, x = −½ |
Roots, Discriminant & Parabola Properties
Quadratic Formula
Solves any quadratic with x = (−b ± √Δ) / 2a, handling whole, decimal, and fractional roots.
Discriminant Analysis
Shows the value of Δ and explains what it means for the number and type of roots.
Vertex & Axis
Displays the parabola's vertex coordinates and its axis of symmetry for graphing.
Complex Roots
When Δ < 0, the solver reports the complex (imaginary) solutions instead of leaving you stuck.
Step-by-Step Working
Numbered cards walk through the discriminant and the formula so you can follow every calculation.
Live Instant Solving
Results and the equation preview update on every keystroke — no submit button to press.
What the Discriminant Tells You
| Discriminant | Roots | Geometry |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses the x-axis at two points. |
| Δ = 0 | One double (repeated) root | Parabola touches the x-axis at its vertex. |
| Δ < 0 | Two complex roots | Parabola never crosses the x-axis. |
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the solutions to any quadratic equation ax² + bx + c = 0 directly from its coefficients, without factoring.
What is the discriminant and what does it tell me?
The discriminant is Δ = b² − 4ac. It determines the nature of the roots: positive means two distinct real roots, zero means one double root, and negative means two complex (non-real) roots.
How do I find the vertex and axis of symmetry?
The vertex is the highest or lowest point of the parabola, located at (−b/2a, f(−b/2a)). It lies on the axis of symmetry, the vertical line x = −b/2a. Both are shown automatically alongside the roots.
Why must coefficient a not be zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0). The quadratic formula divides by 2a, so a non-zero a is required. If you enter a = 0, the solver falls back to solving it as a linear equation.
What happens when there are no real roots?
When Δ < 0, the square root of a negative number gives imaginary values. The solver reports the two complex roots in the form p ± qi so the answer is still complete.
Does it show fractions instead of long decimals?
Yes. When a root is a clean fraction the solver displays it exactly — for example −½ rather than −0.5 — with the decimal value shown in parentheses for reference.
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