What does the discriminant tell me?

The discriminant D = b^2 − 4ac determines the nature of roots: D > 0 two real roots; D = 0 one real repeated root; D < 0 two complex conjugate roots.

Can the calculator show exact roots?

Yes. When inputs are integers, it shows exact forms with square roots when possible and simplifies when the discriminant is a perfect square.

Then the equation is linear (bx + c = 0). The quadratic formula does not apply. Use x = -c/b when b ≠ 0.

What precision is used for decimals?

You can set the decimal precision from 0 to 12 places; the default is 4.

Does the tool support complex roots?

Yes. When D < 0 it outputs complex roots in the form p ± qi and also the exact symbolic form with i.

Quadratic Formula Calculator

Q: What is the quadratic formula?

The quadratic formula x = [-b ± sqrt(b^2 − 4ac)] / (2a) gives the solutions to ax^2 + bx + c = 0 for a ≠ 0.

Q: Can the calculator show exact roots?

Yes. When inputs are integers, it shows exact forms with square roots when possible and simplifies when the discriminant is a perfect square.

Q: What if a = 0?

Then the equation is linear (bx + c = 0). The quadratic formula does not apply. Use x = -c/b when b ≠ 0.

Q: What precision is used for decimals?

You can set the decimal precision from 0 to 12 places; the default is 4.

Q: Does the tool support complex roots?

Yes. When D < 0 it outputs complex roots in the form p ± qi and also the exact symbolic form with i.

A fast, accessible, and authoritative calculator to solve quadratic equations of the form ax² + bx + c = 0. Ideal for students, educators, and professionals who need exact and decimal solutions, discriminant, vertex, axis of symmetry, and an instant plot.

Enter Coefficients

Coefficient a

Coefficient b

Coefficient c

Decimal precision

Exact form Show steps

Results

Equation

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Discriminant (D)

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Nature of roots

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Roots (decimal)

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Roots (exact)

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Vertex (h, k)

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Axis of symmetry

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y-intercept

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Factorization

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Interactive Graph

Data Source and Methodology

Authoritative Data Source: Wolfram MathWorld — Quadratic Equation, Edwin Weisstein, last updated 2023. All calculations are rigorously based on the formulas and definitions provided by this source.

The Formula Explained

Quadratic equation: \( ax^2 + bx + c = 0,\; a \neq 0 \)

Discriminant: \( D = b^2 - 4ac \)

Roots: \( x = \dfrac{-b \pm \sqrt{\,b^2 - 4ac\,}}{2a} \)

Vertex: \( h = -\dfrac{b}{2a},\quad k = a h^2 + b h + c \)

Axis of symmetry: \( x = -\dfrac{b}{2a} \)

Glossary of Variables

a: Quadratic coefficient (must be non-zero).
b: Linear coefficient.
c: Constant term (also equals the y-intercept).
D: Discriminant, D = b² − 4ac, indicating the nature of roots.
x1, x2: The two solutions (may be real or complex).
Vertex (h, k): The turning point of the parabola.
Axis of symmetry: Vertical line x = h passing through the vertex.
Exact form: Symbolic solutions using square roots (and i for complex roots).

How It Works: A Step-by-Step Example

Given a = 1, b = -3, c = -10:

Compute the discriminant: \( D = (-3)^2 - 4(1)(-10) = 9 + 40 = 49 \).
Apply the formula: \( x = \dfrac{-(-3) \pm \sqrt{49}}{2(1)} = \dfrac{3 \pm 7}{2} \).
Compute roots: \( x_1 = \dfrac{10}{2} = 5 \), \( x_2 = \dfrac{-4}{2} = -2 \).
Vertex: \( h = -\dfrac{-3}{2} = 1.5 \), \( k = 1(1.5)^2 - 3(1.5) - 10 = -12.25 \).
Axis of symmetry: \( x = 1.5 \). Factorization: \( (x - 5)(x + 2) \).

Frequently Asked Questions (FAQ)

What is the quadratic formula?

The quadratic formula \( x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives the solutions to any quadratic equation \( ax^2 + bx + c = 0 \) with \( a \neq 0 \).

How do I know how many real roots I have?

Use the discriminant \( D = b^2 - 4ac \): if D > 0 you have two real roots; if D = 0 one real repeated root; if D < 0 two complex conjugate roots.

Can I get exact (symbolic) answers?

Yes. If your inputs are integers, the calculator shows exact forms with square roots and simplifies when the discriminant is a perfect square.

What happens if a = 0?

The equation is linear, not quadratic. Solve using \( x = -c/b \) when \( b \neq 0 \).

How precise are decimal outputs?

You can choose 0–12 decimal places. This only affects decimal approximations; exact forms remain symbolic.

Does the graph update automatically?

Yes, after you calculate, the graph displays the parabola with axes and a symmetric window around the vertex.

Can I share a prefilled link?

Use the Share button to copy a URL with your current inputs, so others can load the same equation instantly.

Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Team.
Last reviewed for accuracy on: September 14, 2025.