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Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula: real and complex roots, discriminant, vertex and axis of symmetry. Free.

1x² + -3x + 2 = 0

Two real roots
x₁ = 2, x₂ = 1
Discriminant
b² − 4ac = 1
Vertex
(1.5, -0.25)
Axis of symmetry
x = 1.5
Parabola opens
Upward
y-intercept
(0, 2)

x = (−b ± √(b² − 4ac)) / 2a

Solving with the quadratic formula

Every quadratic equation ax² + bx + c = 0 can be solved with x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you what kind of answer to expect: positive means two real roots, zero means one repeated root, and negative means two complex roots. Enter a, b and c to get the roots, the discriminant, the vertex of the parabola, its axis of symmetry and its y-intercept.

How to use it

  1. 1

    Write the equation in standard form

    Move everything to one side so it reads ax² + bx + c = 0.

  2. 2

    Enter a, b and c

    Include signs: for x² − 3x + 2 = 0, enter 1, −3 and 2.

  3. 3

    Read the roots

    Plus the discriminant and vertex.

Frequently asked questions

What does the discriminant tell me?

If b² − 4ac > 0 there are two real roots, if it's 0 there's one, and if it's negative the roots are complex numbers.

What if a is 0?

Then it isn't a quadratic but a linear equation, bx + c = 0, with the single solution x = −c/b.

How do I find the vertex?

The x-coordinate is −b/2a; put it back into the equation for the y-coordinate.

Is anything sent to a server?

No. Everything is calculated in your browser, and nothing you type is stored or sent anywhere.

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