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Quadratic Solver online

Solve ax² + bx + c = 0 — step-by-step with real and complex roots

Quadratic Equation Solver logo
by
CHUNKY
MUNSTER
ax² + bx + c = 0

How to Use the Quadratic Equation Solver

  1. Paste or enter your input into the text field.
  2. Configure any options (format, delimiter, encoding, or mode) using the controls above the output.
  3. The result updates instantly — no submit button required for most operations.
  4. Click Copy or Download to take the output to your next step.

Enter the coefficients a, b, and c and the page applies the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant is reported alongside the roots so you can see at a glance whether the solution is two real numbers, one repeated real number, or a complex-conjugate pair.

How the Quadratic Equation Solver Works

When a is zero, the equation collapses to the linear case bx + c = 0 and the solver returns the single root −c/b instead of dividing by zero. Complex roots are shown in standard form as a real part plus or minus an imaginary part. Arithmetic is done in IEEE-754 double precision, which is plenty for textbook problems but can lose accuracy on extremely ill-conditioned cases (very small leading coefficient relative to b).

Frequently Asked Questions

What does the discriminant tell me?

The discriminant Δ = b² − 4ac decides the nature of the roots. Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: a pair of complex-conjugate roots. The sign appears in the result panel before the roots themselves.

Can the solver handle a = 0?

Yes — when a is zero the equation is linear (bx + c = 0) and the tool returns the single root −c/b. If both a and b are zero it reports that there is no equation to solve.

Why are some roots shown as a + bi?

A negative discriminant gives complex roots. The solver returns them as a real part −b/(2a) and an imaginary part √|Δ|/(2a), as a conjugate pair — the standard output of the quadratic formula over the complex numbers.

How precise is the answer?

It is computed in IEEE-754 double precision, which is accurate to about 15 significant figures. For ill-conditioned equations (a very small relative to b), the formula can lose accuracy on one of the two roots — the citardauq form would do better but is overkill for most inputs.

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