What does this tool do?
The quadratic solver solves an equation of the form ax² + bx + c = 0 from its coefficients a, b, and c. It instantly shows the discriminant, the real or complex roots, and the vertex of the parabola. It is ideal for high school and university students, and for anyone who wants a quick read on how a parabola behaves.
How to use it
- Enter your values into the fields.
- The result is calculated instantly.
- Read or copy the result.
How the discriminant decides the roots
The tool first computes the discriminant b² − 4ac and colors the result accordingly. A positive discriminant gives two distinct real roots (green), a value of exactly zero gives one repeated root (orange), and a negative value means there are no real roots, so the tool returns the roots as complex numbers in a + bi form (red). This way you see not only the answer but also the type of solution at a glance.
The vertex and the shape of the parabola
Alongside the roots, the tool reports the vertex of the parabola as (x, y), the minimum or maximum point found at x = −b/2a. The vertex reveals the axis of symmetry and the extreme value of the function. It is directly useful when modelling things like the highest point a thrown object reaches or the lowest value of a cost curve.
Frequently asked questions
What do I see when the roots are complex?
When the discriminant is negative the tool labels the result 'No real roots' and shows the roots as 'real ± imaginary i', for example 2 + 3i and 2 − 3i. The real part comes from −b/2a and the imaginary part from √(−discriminant)/2a.
What happens if I enter 0 for a?
With a = 0 the equation is no longer quadratic but linear, and dividing by 2a becomes undefined. Enter a non-zero value for a to keep a valid parabola.
What does a repeated root mean?
When the discriminant is exactly zero, the parabola touches the x-axis at a single point, so the two roots coincide and return the same value. The tool flags this in orange as 'One repeated root'.
Is Quadratic Equation Solver free?
Yes. Quadratic Equation Solver and all 300+ tools on MagmaNex are completely free with no signup or account required.
Is my data safe?
Yes. Everything runs locally in your browser. The data you enter is never sent to a server, stored, or shared.
Does it work on mobile?
Yes. Quadratic Equation Solver works on phones, tablets and desktops — no app download needed.
🔒 Privacy: everything runs in your browser, your data is never sent to a server.