📉Quadratic Equation Solver
Solve any quadratic equation ax²+bx+c=0. Enter your coefficients for instant real or complex roots, step-by-step working using the quadratic formula, discriminant analysis, and a parabola graph.
Step-by-step working
Parabola Graph
Free Quadratic Equation Solver Online
Stuck on a quadratic equation? Our free Quadratic Equation Solver finds the roots the moment you enter your coefficients. As a fast and reliable quadratic equation solver, it takes any equation in the form ax² + bx + c = 0 and instantly returns the solutions—whether they're real or complex—so you get the answer without the manual arithmetic.
But it does more than spit out a number. See the full step-by-step working using the quadratic formula, get a discriminant analysis that explains how many roots to expect and why, and view a parabola graph that plots your equation and its roots visually. Whether you're checking homework, studying for an exam, or teaching the concept, every part of the solution is laid out clearly.
Simply enter your a, b, and c values to see the solution instantly. No sign-up, no downloads, no limits—just clear answers whenever you need them.
About the Quadratic Equation Solver
Key features
Real and complex roots
Handles all three discriminant cases — two distinct real roots, one repeated root, and two complex (imaginary) roots — displaying each in clear notation.
Step-by-step working
Five labelled steps walk through identifying the coefficients, computing the discriminant, classifying the root type, writing out the quadratic formula, and solving for each root.
Discriminant analysis
The discriminant (Δ = b² − 4ac) is calculated and colour-coded — green for two real roots, amber for one repeated root, and violet for complex roots — so the root type is immediately obvious.
Parabola graph
An SVG graph plots the parabola y = ax² + bx + c, marks the vertex, and marks the real roots where the curve crosses the x-axis — or shows a floating parabola when roots are complex.
Who it's for
Students
Check homework answers and follow the step-by-step working to understand where you went wrong or confirm your method is right.
Teachers
Project the solver on screen to walk through examples live. Change coefficients and immediately see how the roots, discriminant, and graph respond.
Exam candidates
Use it to verify answers quickly when practising past papers. The discriminant display confirms at a glance whether the expected number of roots is correct.
Curious learners
Experiment with different coefficients to see how changing a, b, and c shifts the parabola and affects whether the roots are real or complex.
How the calculation works
The solver applies the standard quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac is computed first and compared against zero (with a small floating-point tolerance for near-zero values) to determine which case applies.
For positive discriminants, both roots are real and distinct: x₁ = (−b + √Δ) / (2a) and x₂ = (−b − √Δ) / (2a). For a zero discriminant, the single repeated root is x = −b / (2a), which equals the x-coordinate of the vertex. For a negative discriminant, the roots are complex conjugates: x = −b/(2a) ± i·√(−Δ)/(2a), where the real part is the vertex x-coordinate and the imaginary part is derived from the square root of the absolute discriminant.
The parabola graph plots y = ax² + bx + c over a range centred on the vertex, scaled automatically so both roots (or the vertex region for complex roots) are clearly visible.
Frequently Asked Questions
How does the quadratic equation solver work?
Enter the coefficients a, b, and c from your equation (ax² + bx + c = 0) and the tool instantly calculates the roots using the quadratic formula, along with step-by-step working — no button refresh needed.
Is this quadratic solver free to use?
Yes, it's completely free with no sign-up, downloads, or limits. Solve as many equations as you need, as often as you like.
Does it show the steps?
Yes. The solver shows the full step-by-step solution using the quadratic formula, so you can follow exactly how the answer is reached — identifying the coefficients, calculating the discriminant, classifying the root type, and applying the formula to get each root. Not just the final result.
Can it solve equations with complex roots?
Yes. When the discriminant is negative, the solver returns the complex (imaginary) roots in a + bi form rather than just saying there is no real solution. The step-by-step working shows exactly how the imaginary parts are derived.
What is the discriminant and why does it matter?
The discriminant (Δ = b² − 4ac) tells you how many roots an equation has and what type they are. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are two complex roots. The tool calculates and displays the discriminant prominently so you always know what to expect before reading the roots.
Does it graph the equation?
Yes. The solver plots the parabola for your equation, marks the roots with dots on the x-axis (for real roots), and highlights the vertex. This makes it easy to visualise whether the parabola crosses the x-axis, just touches it, or floats entirely above or below it.
What if a is 0?
If a is 0 the equation is not quadratic — without an x² term it becomes a linear equation (bx + c = 0). The tool will display a clear error message rather than attempting to calculate an invalid result. Enter any non-zero value for a to use the quadratic solver.
Is my data stored anywhere?
No. Your inputs are processed directly in your browser and are never stored or sent to a server, keeping your work completely private. Close the tab and everything is gone.
Does it work on mobile devices?
Yes, the tool is fully responsive and works on smartphones, tablets, and desktops without any installation. The input fields, step-by-step working, and graph all adapt to the screen width.
How accurate are the results?
Calculations use precise floating-point math, giving accurate roots, discriminant values, and graph points, including for complex and fractional results. For near-zero discriminants the solver applies a small tolerance to correctly classify them as the zero case rather than reporting spurious tiny imaginary parts.