The equation is linear, not quadratic, and has the single solution x = −c ÷ b.

What does a complex root mean in practice?

In a physical model it usually means no real solution exists — a projectile that never reaches the given height, for instance.

Can I factorise instead?

Yes, when the roots are simple rationals. The formula always works, which is why it is the reliable fallback.

Quadratic Equation Calculator

Solve any quadratic equation using the quadratic formula, including complex roots, and see the discriminant and vertex that describe the shape of the parabola.

Fill in the fields above and select Calculate to see your result and the full working.

Formula used

x = (−b ± √(b² − 4ac)) ÷ 2a

How to use this calculator

Rearrange your equation into the form ax² + bx + c = 0 first.
Enter the three coefficients, including any minus signs.
The discriminant tells you what kind of roots to expect before you read them.

Example calculation

Solve x² − 3x + 2 = 0:

Discriminant = 9 − 8 = 1

x = (3 ± 1) ÷ 2

x = 2 or x = 1.

What does this result mean?

The discriminant b² − 4ac determines everything. Positive means the parabola crosses the x-axis twice; zero means it touches at the vertex; negative means it never meets the axis and the roots are complex conjugates. The vertex at x = −b ÷ 2a is the minimum when a is positive and the maximum when a is negative.

Frequently asked questions

What if a is zero?: The equation is linear, not quadratic, and has the single solution x = −c ÷ b.
What does a complex root mean in practice?: In a physical model it usually means no real solution exists — a projectile that never reaches the given height, for instance.
Can I factorise instead?: Yes, when the roots are simple rationals. The formula always works, which is why it is the reliable fallback.

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