About the Quadratic Equation Solver
A free, instant solver that finds the roots of any quadratic equation and shows the complete working.
What it is
The Quadratic Equation Solver solves equations of the form ax² + bx + c = 0 by applying the quadratic formula. Enter your three coefficients and results update instantly — no submit button, no page reload.
Everything runs in your browser. Your inputs are never sent to a server, stored, or logged. Close the tab and they are gone.
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.
Completely free
No sign-up, no rate limit, no premium tier. Solve equations with integer, decimal, or fractional coefficients — the solver handles them all instantly. If you find it useful, sharing it with a classmate is all we ask.