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Quadratic Formula Calculator

Solve ax² + bx + c = 0, classify the discriminant, and find real or complex roots, the vertex, axis of symmetry, and parabola direction.

  • quadratic equation solver
  • quadratic roots
  • discriminant
  • parabola vertex

The non-zero coefficient of x².

The coefficient of x.

The constant term.

Status: initial

Results

Awaiting calculation

Calculator guide

Solve any quadratic in standard form

Enter the coefficients of ax² + bx + c = 0 to calculate the discriminant, real or complex roots, vertex, axis of symmetry, and the direction in which the parabola opens.

When to use the quadratic formula

Use the quadratic formula when factoring is difficult or impossible over the integers. It solves every quadratic with a non-zero leading coefficient and makes the number and type of roots visible through the discriminant.

How the roots and graph features are found

The discriminant D = b² − 4ac determines whether roots are distinct real, repeated real, or complex. Substituting D into x = (−b ± √D)/(2a) gives the roots. The vertex lies on x = −b/(2a).

Variable explanations

Understand what each input and result means before calculating.

Coefficient a

The non-zero multiplier of x²; its sign controls whether the parabola opens up or down.

Coefficient b

The multiplier of x that affects the axis of symmetry and roots.

Constant c

The y-intercept of the quadratic function.

Discriminant

The expression b² − 4ac that classifies the roots.

Complex root

A root containing the imaginary unit i when the discriminant is negative.

Reviewed by the Calculator.org.in Editorial Team

Formula behavior, validation cases, explanatory examples, and cited sources are checked before publication. This review supports educational accuracy and is not a substitute for qualified professional advice.

Last reviewed: 2026-08-08

Review process

Formula guide

See the calculation logic, variable definitions, and practical meaning.

Quadratic roots

x = (−b ± √(b² − 4ac)) / 2a

  • a must be non-zero
  • b and c may be any real numbers

The plus and minus signs produce the two solutions.

Discriminant

D = b² − 4ac

  • D > 0: two real roots
  • D = 0: repeated root
  • D < 0: complex pair

The discriminant classifies solutions before square roots are evaluated.

Vertex

h = −b/(2a); k = ah² + bh + c

  • Vertex = (h, k)

The axis of symmetry is the vertical line x = h.

Worked examples

Follow realistic inputs through the calculation step by step.

1

Worked example

Two real roots

  1. 1For x² − 5x + 6 = 0, use a = 1, b = −5, c = 6.
  2. 2D = 25 − 24 = 1.
  3. 3The roots are 3 and 2.
2

Worked example

One repeated root

  1. 1For x² + 2x + 1 = 0, D = 0.
  2. 2Both formula branches give x = −1.
  3. 3The parabola touches the x-axis at its vertex.
3

Worked example

Complex roots

  1. 1For x² + 1 = 0, use a = 1, b = 0, c = 1.
  2. 2D = −4.
  3. 3The roots are i and −i.

Common mistakes

Avoid these common input and interpretation errors.

Not using standard form

Move every term to one side so the other side equals zero before identifying coefficients.

Dropping the sign of b

If the equation contains −5x, enter b = −5.

Forgetting the entire denominator

Both −b and ±√D are divided by 2a.

Treating a negative discriminant as no solution

It means no real roots, but two complex conjugate roots still exist.

Frequently asked questions

Quick answers to the questions users ask most often.

What if coefficient a is zero?
The equation is linear rather than quadratic, so the quadratic formula does not apply.
What does the discriminant tell me?
It tells whether the quadratic has two distinct real roots, one repeated real root, or two complex roots.
Can the calculator show complex roots?
Yes. Negative discriminants return roots in a ± bi form.
Are the roots the x-intercepts?
Real roots are x-intercepts of the graph. Complex roots do not appear as intercepts on the real coordinate plane.
How is the vertex related to the roots?
The vertex lies on the axis of symmetry, which is halfway between two real roots when both exist.
Can every quadratic be factored easily?
No. The quadratic formula works even when factors are irrational or complex.

References

Sources used to support the calculator guidance.

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Version history

A transparent record of calculator content updates.

Updated 2026-08-08
  • 1.0.0 · 2026-08-08

    Initial release with real and complex roots, discriminant classification, vertex, axis, and worked steps.