What is the quadratic formula?

For an equation ax² + bx + c = 0 (with a ≠ 0), the quadratic formula gives the roots as x = (-b ± √(b² - 4ac)) / 2a. It always works, regardless of whether the roots are real or complex.

What does the discriminant tell me?

The discriminant is b² - 4ac. If it's positive, the equation has two distinct real roots. If it's zero, there's exactly one repeated real root. If it's negative, the roots are a complex conjugate pair.

What happens if I enter 0 for coefficient a?

If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0). The calculator will show an error since the quadratic formula requires a ≠ 0.

Can this calculator handle complex roots?

Yes. When the discriminant is negative, the calculator returns both complex roots in the form p ± qi, fully simplified.

What is the vertex of a parabola and why does it matter?

The vertex is the highest or lowest point of the parabola y = ax² + bx + c, located at x = -b/2a. It tells you the equation's maximum or minimum value, which is useful for optimization problems.

What is the axis of symmetry?

The axis of symmetry is the vertical line x = -b/2a that divides the parabola into two mirror-image halves. It always passes through the vertex.

Can I use decimal or negative coefficients?

Yes, a, b, and c can be any real numbers, including decimals and negative values. The calculator handles all valid combinations correctly.

How is this different from factoring?

Factoring only works cleanly when the roots are rational numbers, which is often not the case. The quadratic formula works for every quadratic equation, including ones with irrational or complex roots.

Quadratic Formula Calculator — Solve Any Quadratic Equation

Our Quadratic Formula Calculator solves any equation in the form ax² + bx + c = 0, showing the full step-by-step substitution into x = (-b ± √(b² - 4ac)) / 2a. You'll see the discriminant, both roots (real or complex), the vertex of the parabola, and the axis of symmetry — everything you need to fully understand and graph the equation, not just the final answer.

Quick Answer

The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, solves any equation of the form ax² + bx + c = 0. This free calculator shows the discriminant, both roots (real or complex), the vertex, and the axis of symmetry with full step-by-step working.

Solve ax² + bx + c = 0

Enter coefficients a, b, and c, then click Solve.

How to Use the Quadratic Formula Calculator — Solve ax² + bx + c = 0

1
Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
2
Click 'Solve' to calculate the roots using the quadratic formula.
3
Review the discriminant to see whether the roots are two real numbers, one repeated real number, or a complex pair.
4
Check the step-by-step substitution to see exactly how each root was calculated.
5
View the vertex and axis of symmetry to understand the parabola's shape without graphing it by hand.

Why Use Quadratic Formula Calculator — Solve ax² + bx + c = 0?

The quadratic formula is one of the most frequently used tools in algebra, but a single sign error in b² - 4ac or in the ± step produces a wrong answer that looks plausible. Our calculator performs the substitution exactly and shows every intermediate step, so you can check your own work or get a trustworthy answer instantly. It also classifies the discriminant automatically, which is where most students lose points — correctly identifying two real roots, a repeated root, or a complex conjugate pair takes understanding, not just computation.

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