Quadratic Formula Calculator

Use this free quadratic formula calculator to solve ax^2 + bx + c = 0, find real or complex roots, discriminant, vertex, axis of symmetry, steps, copy, and history.

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Real and complex roots Discriminant Vertex and axis Step-by-step work

Standard formx^2 - 3x + 2 = 0

Coefficient aCoefficient bCoefficient c

The discriminant is positive, so the equation has two real roots.x = 2, 1

Discriminant: 1
Vertex: (1.5, -0.25)
Axis: x = 1.5

Steps

Start with standard form: x^2 - 3x + 2 = 0.
Use a = 1, b = -3, and c = 2.
Find the discriminant: b^2 - 4ac = (-3)^2 - 4(1)(2) = 1.
Use the quadratic formula: x = (-b +/- sqrt(discriminant)) / 2a, with 2a = 2.
The roots are x = 2, 1.

How to use the Quadratic Formula Calculator

Enter coefficients a, b, and c from standard form ax^2 + bx + c = 0.
Press Calculate roots to use the quadratic formula.
Review the roots, discriminant, vertex, axis of symmetry, and step-by-step work.
Use examples, recent answers, or copy the answer while you study or graph.

What people use it for

Solve quadratic equations in standard form ax^2 + bx + c = 0.

Check whether an equation has two real roots, one repeated root, or complex roots.

Find the discriminant, vertex, axis of symmetry, opening direction, and y-intercept.

Copy roots and steps for algebra homework, graphing, studying, or checking work.

Quick examples

Two real roots

x^2 - 3x + 2 = 0

x = 2, 1

Repeated root

x^2 - 4x + 4 = 0

x = 2

Complex roots

x^2 + 2x + 5 = 0

x = -1 +/- 2i

Need the guide or a nearby tool?

Need a slower walkthrough, a related calculator, or the full library? These links keep you close to the task you started.

How to use the Quadratic Formula Calculator See slower examples, input meanings, formula notes, result limits, and mistakes to avoid.
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Frequently asked questions

Quick answers about the quadratic formula, discriminants, complex roots, graph details, coefficients, and privacy.

What formula does the Quadratic Formula Calculator use?

It uses x = (-b +/- sqrt(b^2 - 4ac)) / 2a for equations written in standard form ax^2 + bx + c = 0.

What is the discriminant?

The discriminant is b^2 - 4ac. It tells you the root type: positive means two real roots, zero means one repeated real root, and negative means two complex conjugate roots.

What do the main Quadratic Formula Calculator inputs mean?

The main inputs are the numbers, operation, mode, or known values the calculator needs. Keep units consistent, enter percentages the way the page label shows, and use the examples as a quick check before trusting the answer.

How should I read the Quadratic Formula Calculator answer?

Read the headline answer, then check the supporting lines and examples to understand how the calculator got there. If one input changes, rerun the tool and compare the new answer instead of guessing.

What should I double-check before trusting the Quadratic Formula Calculator?

Check units, signs, rounding, and the selected mode before copying the answer. If the number feels weird, rerun one of the examples first, then put your own values back in slowly.

Why can coefficient a not be zero?

A quadratic equation needs an x^2 term. If a is zero, the equation becomes linear, so the quadratic formula does not apply.

Can this calculator show complex roots?

Yes. When the discriminant is negative, the calculator shows the complex conjugate roots using i.

Does this calculator show graph details?

Yes. It shows the vertex, axis of symmetry, y-intercept, and whether the parabola opens up or down.

What form should I enter the equation in?

Enter the coefficients from standard form ax^2 + bx + c = 0. For example, x^2 - 3x + 2 = 0 uses a = 1, b = -3, and c = 2.

Is my quadratic calculation history private?

Yes. Recent quadratic answers stay only in the current browser tab while you use the page. They are not sent to a server.