What does the discriminant tell me?

The discriminant is b² − 4ac. If it is positive there are two distinct real roots, if it is zero there is one repeated real root, and if it is negative there are two complex (imaginary) roots.

If a is zero the equation is no longer quadratic — it is linear — so the quadratic formula does not apply. The solver shows a note asking for a non-zero a value.

How are complex roots shown?

When the discriminant is negative the roots are written as p ± qi, where p is −b÷2a (the real part) and q is √(−discriminant)÷2a (the imaginary part). For example x = 1 ± 2i.

How does the solver handle decimals and rounding?

Coefficients can be any decimal, and results are rounded to six decimal places for readability, so a clean answer like 2 shows as 2 rather than 2.000000.

Can a, b or c be negative?

Yes. Enter negatives directly, for example a = 1, b = −3, c = 2. The signs are carried through the formula exactly as written.

What is the quadratic formula itself?

x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two roots, the square root term is the discriminant, and dividing by 2a scales them to your specific equation.

Is anything sent to a server?

No — the quadratic formula is applied entirely in your browser, and the coefficients you enter are never uploaded.

Quadratic Formula Solver

Quadratic formula solver

This tool solves any equation of the form ax² + bx + c = 0 for x. Enter the three coefficients a, b and c and it instantly returns the roots — whether they are two real numbers, a single repeated value, or a complex conjugate pair — along with the discriminant so you can see why. It is useful for algebra homework, physics problems (projectile and motion equations), and any time you need to factor or find where a parabola crosses zero.

How it works

The solver applies the quadratic formula:

x = (−b ± √(b² − 4ac)) / 2a

First it computes the discriminant, D = b² − 4ac, which decides the kind of answer:

D > 0 — two distinct real roots: x₁ = (−b + √D)/2a and x₂ = (−b − √D)/2a.
D = 0 — one repeated real root: x = −b/2a.
D < 0 — two complex roots: x = (−b/2a) ± (√(−D)/2a)·i.

If a = 0 the equation is linear, not quadratic, so the formula does not apply and the tool prompts for a non-zero a. Results are rounded to six decimal places.

Example

For x² − 3x + 2 = 0 (a = 1, b = −3, c = 2):

Discriminant D = (−3)² − 4·1·2 = 9 − 8 = 1 (positive, so two real roots).
x₁ = (3 + √1) / 2 = 4 / 2 = 2
x₂ = (3 − √1) / 2 = 2 / 2 = 1

Discriminant b² − 4ac	Root type	Result shape
Positive	Two real roots	x₁, x₂
Zero	One repeated root	x
Negative	Two complex roots	p ± qi

Everything runs in your browser, so the coefficients you enter are never uploaded.