Quadratic Equation Calculator
Enter the coefficients a, b, and c to solve ax² + bx + c = 0 with the quadratic formula — get the real or complex roots, discriminant, vertex, and axis of symmetry, with every step shown.
- 01Solve any quadratic equation ax² + bx + c = 0 instantly with the quadratic formula.
- 02Get both real roots, a repeated root, or a complex conjugate pair automatically.
- 03See the discriminant b² − 4ac and what it tells you about the nature of the roots.
- 04Find the vertex of the parabola and the axis of symmetry in one click.
- 05See step-by-step working that substitutes your values into the quadratic formula.
- 06100% free and private — every calculation runs in your browser.
Quadratic Equation Calculator
Enter the coefficients of ax² + bx + c = 0
1x² + −3x + 2 = 0
Try an example equation
Root x₁
2
Real solution of ax² + bx + c = 0
Root x₂
1
Real solution of ax² + bx + c = 0
Discriminant (b² − 4ac)
1
Nature of Roots
Two distinct real roots
Axis of Symmetry
x = 1.5
Vertex (h, k)
(1.5, −0.25)
Vertex x (h)
1.5
Vertex y (k)
−0.25
Coefficient a
1
Coefficient b
−3
Step-by-step calculation
- 01Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a
- 02Substitute the values: x = (−(−3) ± √((−3)² − 4 · 1 · 2)) ÷ (2 · 1)
- 03Discriminant b² − 4ac = (−3)² − 4 · 1 · 2 = 1
- 04Two real roots: x₁ = 2, x₂ = 1
Why Use This Quadratic Equation Calculator
Real and Complex Roots
The calculator automatically detects whether your quadratic equation has two distinct real roots, one repeated real root, or a complex conjugate pair, and formats complex roots cleanly as p ± q i. You always get the correct roots of the quadratic equation, whatever the discriminant.
Step-by-Step Quadratic Formula
See exactly how the quadratic formula is applied: the formula itself, your values for a, b, and c substituted in, the discriminant b² − 4ac, and the final roots. Perfect for homework, exam revision, and checking your own work when you solve a quadratic equation.
Discriminant and Nature of Roots
The discriminant calculator computes b² − 4ac and tells you what it means: greater than zero gives two real roots, equal to zero gives one repeated root, and less than zero gives complex roots. Understanding the discriminant is key to mastering quadratic equations.
Vertex and Axis of Symmetry
Beyond the roots, the calculator finds the vertex of the parabola (h, k) and the axis of symmetry x = −b ÷ 2a, so you can sketch the graph of y = ax² + bx + c and understand its shape at a glance.
Instant and Private
Everything runs entirely in your browser with no server round-trips. Your equation never leaves your device, results appear instantly as a quadratic equation solver should, and there is no signup or installation required.
Free with No Limits
Solve as many quadratic equations as you need — no daily limits, no account, and no paywall. The full quadratic formula calculator is completely free.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. The values of x that satisfy the equation are called the roots or solutions, and a quadratic always has exactly two roots (which may be equal, or may be complex). Quadratic equations appear everywhere in algebra, physics, engineering, and economics.
Whether you are a student learning how to solve quadratic equations, a teacher preparing examples, or an engineer checking a calculation, this quadratic equation calculator gives instant, step-by-step results with the quadratic formula for any values of a, b, and c.
- The Quadratic Formula
- The quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a solves any quadratic equation directly from its coefficients. This calculator uses the quadratic formula and shows each substitution, so you can follow exactly how the roots of the quadratic equation are found.
- The Discriminant
- The discriminant is the part under the square root, b² − 4ac. It decides the nature of the roots: if it is positive there are two distinct real roots, if it is zero there is one repeated real root, and if it is negative there are two complex conjugate roots. A discriminant calculator is the quickest way to classify a quadratic before solving it.
- Real vs Complex Roots
- When the discriminant is negative the square root of a negative number is imaginary, so the roots are complex numbers written as p ± q i, where the real part p equals −b ÷ 2a and the imaginary magnitude q equals √(−(b² − 4ac)) ÷ 2a. The calculator formats these conjugate roots for you automatically.
- Vertex, Axis of Symmetry, and the Parabola
- The graph of a quadratic is a parabola. Its vertex sits at x = −b ÷ 2a with y = c − b² ÷ 4a, and the axis of symmetry is the vertical line x = −b ÷ 2a through that vertex. Knowing how to solve quadratic equations together with the vertex makes graphing and optimisation problems much easier.
How to Use the Quadratic Equation Calculator
- 01
Write your equation in standard form
Rearrange your equation so that everything is on one side and it reads ax² + bx + c = 0. For example, x² = 3x − 2 becomes x² − 3x + 2 = 0, giving a = 1, b = −3, and c = 2. Remember that a must not be zero for the equation to be quadratic.
- 02
Enter the coefficients a, b, and c
Type the coefficient a (the number in front of x²), b (the number in front of x), and the constant c into the three input boxes. Decimals and negative numbers are supported, so any quadratic equation works.
- 03
Click Solve
Press the Solve button. The quadratic formula calculator computes the discriminant, applies the quadratic formula, and returns the roots — real or complex — instantly in your browser.
- 04
Read the roots and the steps
The headline cards show the roots of the quadratic equation. Below them you get the discriminant, the nature of the roots, the vertex, and the axis of symmetry, plus a step-by-step section showing your values substituted into the quadratic formula. Great for learning and for checking homework.
Tips for Solving Quadratic Equations
Always Use Standard Form
Before reading off a, b, and c, move every term to one side so the equation equals zero. Mixing up the signs of the coefficients is the most common mistake when people solve a quadratic equation by the quadratic formula.
Check the Discriminant First
Compute b² − 4ac before anything else. The sign of the discriminant tells you immediately whether to expect two real roots, one repeated root, or a complex conjugate pair, which helps you sanity-check the final answer.
Mind the Sign of b
The quadratic formula starts with −b, so a negative b becomes positive inside the formula. Wrapping b in brackets, as in −(−3), prevents sign errors when you substitute values into the quadratic formula.
Try Factoring When It Is Easy
If the quadratic factors neatly, factoring can be faster than the formula. The quadratic formula always works, though, so use this calculator to confirm the roots of the quadratic equation you found by factoring.
Keep Enough Precision
Avoid rounding the discriminant or its square root too early — rounding before the final step can introduce noticeable errors. Round only the final roots for reporting.
Use the Vertex to Graph
Once you have the roots and the vertex, you can sketch the parabola quickly: the axis of symmetry passes through the vertex and the roots are where the curve crosses the x-axis.
Quadratic Formula Definitions and Key Formulas
Definition of a quadratic equation
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0 with a ≠ 0. Its solutions, called roots, are the x-values where the parabola y = ax² + bx + c crosses the x-axis.
What the discriminant tells you
- Discriminant > 0: two distinct real roots (the parabola crosses the x-axis twice).
- Discriminant = 0: one repeated real root (the parabola touches the x-axis at the vertex).
- Discriminant < 0: two complex conjugate roots (the parabola does not cross the x-axis).
- The discriminant b² − 4ac is the fastest way to classify the roots of a quadratic equation.
Ways to solve a quadratic equation
You can solve a quadratic equation by factoring, by completing the square, or by the quadratic formula. The quadratic formula always works for any a, b, and c, which is why this calculator uses it.
Key Quadratic Formulas
Standard form
ax² + bx + c = 0, a ≠ 0
Example: x² − 3x + 2 = 0 has a = 1, b = −3, c = 2.
Quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Gives both roots directly from the coefficients.
Discriminant
Δ = b² − 4ac
Decides whether the roots are real, repeated, or complex.
Complex roots
x = (−b ÷ 2a) ± (√(−(b² − 4ac)) ÷ 2a) i
Used when the discriminant is negative; the roots are conjugates.
Vertex of the parabola
(−b ÷ 2a, c − b² ÷ 4a)
The turning point of y = ax² + bx + c.
Axis of symmetry
x = −b ÷ 2a
The vertical line through the vertex.
Quadratic Equation Calculator FAQ
Q01How do I solve a quadratic equation?
Write the equation in standard form ax² + bx + c = 0, then apply the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. Compute the discriminant b² − 4ac, take its square root, and evaluate the two values of x. This calculator does all of these steps for you, applies the quadratic formula, and shows the working so you can learn how to solve quadratic equations.
Q02What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both roots of any quadratic equation ax² + bx + c = 0 directly from the coefficients a, b, and c, where a is not zero. The ± symbol means you evaluate it once with a plus and once with a minus to get the two roots.
Q03What is the discriminant and what does it tell me?
The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. If it is positive the equation has two distinct real roots, if it is zero it has one repeated real root, and if it is negative it has two complex conjugate roots. A discriminant calculator classifies the roots before you solve the equation.
Q04What are complex roots of a quadratic equation?
When the discriminant b² − 4ac is negative, the square root is imaginary, so the roots are complex numbers written as p ± q i. The real part p equals −b ÷ 2a and the imaginary magnitude q equals √(−(b² − 4ac)) ÷ 2a. The two complex roots are conjugates of each other, and this calculator formats them automatically.
Q05Why must a not be zero?
If a equals zero there is no x² term, so the equation ax² + bx + c = 0 collapses to the linear equation bx + c = 0, which is not quadratic. The quadratic formula divides by 2a, so a = 0 is undefined. This calculator returns an error when a is zero.
Q06How do I find the vertex and axis of symmetry?
The vertex of the parabola y = ax² + bx + c is at x = −b ÷ 2a, and you find its y-coordinate by substituting that x back in, which gives y = c − b² ÷ 4a. The axis of symmetry is the vertical line x = −b ÷ 2a through the vertex. The calculator reports both for every equation.
Q07Can I enter decimals and negative coefficients?
Yes. You can enter integers, decimals, and negative numbers for a, b, and c. The calculator parses all of them, so any quadratic equation — including ones with fractional or negative coefficients — can be solved with the quadratic formula.
Q08Is my data sent to a server?
No. The quadratic equation solver runs entirely in your browser using JavaScript. Your coefficients are never uploaded or stored anywhere, so it is safe to use for any homework or work problem.
Q09Is this quadratic equation calculator free?
Yes, it is completely free with no limits, no signup, and no premium tier. Solve as many quadratic equations as you like with the quadratic formula calculator.