Math calculator

Free quadratic formula calculator

Solve any quadratic equation ax² + bx + c = 0 — enter a, b, and c to find both roots using the quadratic formula, updated live, as you type.

On this page10 sections

InputsLive

a (x² coefficient)

Must not be zero.

b (x coefficient)

c (constant)

Result

Roots

x₁=3 x₂=2

1x² + (-5)x + (6) = 0

Discriminant (Δ)1

x₁3

x₂2

Real roots2

x = (−b ± √(b²−4ac)) / 2a. Δ < 0 → no real roots.

Results are estimates. Consult a professional.

How it's calculated

How the quadratic formula works

The quadratic formula solves any equation of the form ax²+bx+c=0, giving the exact values of x where the parabola crosses the x-axis. Plug in the three coefficients a, b, and c, and the formula does the rest.

x = (−b ± √(b² − 4ac)) / 2a

Discriminant: Δ = b² − 4ac

Weisstein, E. W. Quadratic Formula. MathWorld.

Example

Worked example: solving 2x²−4x−6=0

Example: 2x²−4x−6=0 (a=2, b=−4, c=−6)

Identify the coefficients: a = 2, b = −4, c = −6. Substitute into the formula and simplify step by step.

Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64

x = (4 ± √64) / 4 = (4 ± 8) / 4

x₁ = (4 + 8) / 4 = 3

x₂ = (4 − 8) / 4 = −1

x = 3 or −1

The parabola crosses the x-axis at x = 3 and x = −1.

Quick reference

Discriminant interpretation

The discriminant (Δ = b²−4ac) tells you how many real roots exist before you finish the calculation.

Discriminant (Δ)	Number of real roots	Description
Δ > 0	2 distinct roots	Parabola crosses x-axis twice
Δ = 0	1 repeated root	Parabola touches x-axis at vertex
Δ < 0	0 real roots	Parabola does not cross x-axis (complex roots)

Source: standard algebra curriculum.

Practical tips

Tips for using the quadratic formula

A few habits will prevent the most common errors when applying the quadratic formula.

Rearrange first — move all terms to one side so the equation is in the form ax²+bx+c=0 before identifying a, b, and c.
Watch negative signs — if b is negative, −b becomes positive. Write out the substitution explicitly to avoid sign errors.
Check the discriminant first — computing Δ before the full formula tells you immediately whether real solutions exist.
Simplify the square root — factor out perfect squares from √Δ to get a cleaner answer (e.g. √48 = 4√3).
Verify by substitution — plug each root back into the original equation to confirm both sides equal zero.

Accuracy & limits

Accuracy and limitations

The quadratic formula is exact for any quadratic with rational or irrational coefficients. Results displayed as decimals are limited by floating-point precision (typically 15 significant digits). The formula only applies to degree-2 polynomials — for higher degrees, different methods (cubic formula, numerical solvers) are required. When the discriminant is negative, roots are complex numbers of the form p ± qi and cannot be plotted on a standard number line.

Glossary

Key terms

A numerical factor in a term — a, b, and c in ax²+bx+c=0.

The expression b²−4ac that determines the number and type of roots.

A value of x that satisfies the equation (also called a solution or zero).

The U-shaped graph of a quadratic function y = ax²+bx+c.

A root containing the imaginary unit i = √(−1), occurring when Δ < 0.

The algebraic technique used to derive the quadratic formula from ax²+bx+c=0.

About

About this calculator

Part of our math calculators suite — explore all calculators.

Questions

Frequently asked questions about the free quadratic formula calculator

A quadratic formula calculator is a free online tool that helps you solve any quadratic equation ax² + bx + c = 0 — finds both roots and the discriminant instantly. The quadratic formula gives the exact roots of any quadratic equation in standard form. The discriminant (b² − 4ac) tells you how many real solutions exist before you compute them. It runs entirely in your browser with instant results and no sign-up.

The discriminant is the part under the square root: Δ = b² − 4ac. It tells you the nature of the roots BEFORE you compute them. Δ > 0 means two distinct real roots; Δ = 0 means one repeated real root (the parabola is tangent to the x-axis); Δ < 0 means no real roots (the parabola never touches the x-axis).

Then it's not a quadratic — it's a linear equation, bx + c = 0, which has a single solution x = −c/b. The quadratic formula is undefined in this case.

When Δ < 0, the roots are complex conjugates of the form (−b ± i√|Δ|) / 2a. This calculator returns only real results; for complex roots, you'd need a calculator that handles imaginary numbers.