Quadratic Equation Solver
Solve ax²+bx+c=0 with steps, roots and vertex · free, no signup
Free Quadratic Equation Solver — Roots, Discriminant and Vertex
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Its solutions (roots) are given by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The part under the root, b² − 4ac, is the discriminant, and it tells you whether the roots are two real numbers, one repeated real number or a complex pair.
Enter the coefficients a, b and c and this solver shows the discriminant, both roots (including complex roots written with i), the vertex and axis of symmetry of the parabola, the y-intercept and the factored form where it exists — with the working written out.
Key features
Real and complex roots
Handles every case, including negative discriminants.
Parabola details
Vertex, axis of symmetry and y-intercept.
Factored form
Shown whenever the roots are real.
Step-by-step
The quadratic formula filled in with your numbers.
How to use it
- Enter the coefficients a, b and c.
- Read the discriminant and the roots.
- Check the vertex and factored form.
- Copy the solution.
Worked example
Example
x² − 5x + 6 = 0 → discriminant 1 → x = 2 or x = 3, factored (x − 2)(x − 3) x² + 2x + 5 = 0 → discriminant −16 → x = −1 ± 2i
Who uses this tool
Students
Check algebra homework and see the steps.
Engineers and scientists
Solve projectile and optimisation equations quickly.
Teachers
Prepare worked examples and answer keys.
Tips for the best results
- If the discriminant is a perfect square the equation factors neatly.
- Always check that a is not zero.
- Use the vertex to find the maximum or minimum value.
- Substitute the roots back into the equation to verify.
Common mistakes to avoid
- Dropping the sign of b in the formula.
- Forgetting the ± which gives two roots.
- Treating a negative discriminant as “no solution” — it has complex solutions.
Why use AZRS QuickFix?
It is 100% free, needs no signup and has no watermark or usage limits. The tool runs in your browser, so what you type stays on your device, and it works on phones, tablets and desktops. New tools are added every week — bookmark this page or browse the full QuickFix toolbox.
Frequently asked questions
What is the discriminant?
b² − 4ac. Positive means two real roots, zero means one repeated root, negative means complex roots.
What if a = 0?
The equation is linear, not quadratic; the tool will tell you and solve it as linear.
What are complex roots?
Roots involving i, the square root of −1, which occur when the parabola does not cross the x-axis.
How do I find the vertex?
The x-coordinate is −b / 2a; substitute it to find y.
Can I use decimals and fractions?
Yes, enter them as decimals.
Is it free?
Yes.