Are complex roots “wrong”?

No—they are valid solutions over the complex numbers. They are simply not real x-intercepts.

Does a negative discriminant mean no solution?

It means no real solution. The worksheet still reports complex conjugates.

Is the vertex a root?

Only when D = 0. Otherwise the vertex is the turning point of the parabola, not necessarily an intercept.

Quadratic Equation Solver

Q: What if a is zero?

Then the equation is linear, not quadratic. This solver requires a ≠ 0.

Discriminant-first roots for ax² + bx + c = 0—real when possible, complex when necessary, always labeled.

Real & complex

Discriminant check

Vertex included

Applies the quadratic formula, reports discriminant status, labels complex roots explicitly when D < 0, and includes the vertex (−b/2a, f(vertex)).

Limitation: Requires a ≠ 0.

Jump to calculator · Sources & review

a (coeff of x²)b (coeff of x)c (constant)

How to Solve a Quadratic

Write the equation in standard form ax² + bx + c = 0, then enter a, b, and c.

Confirm the equation is quadratic (a ≠ 0).

Move everything to one side

If you start from 2x² = 5x − 3, rewrite as 2x² − 5x + 3 = 0 before entering coefficients.

Reading the Discriminant

D > 0 yields two distinct real roots, D = 0 a repeated real root, and D < 0 a complex conjugate pair. The worksheet also reports the parabola vertex.

Worked Quadratic Examples

Two real roots

Coefficients

a:1

b:-5

c:6

Roots

x = 3 and x = 2

D = 1 → (−(−5) ± 1) / 2 → 3 and 2.

Complex conjugates

Coefficients

a:1

b:0

c:1

Roots

i and −i (complex)

D = −4 → no real x-intercepts; roots are labeled complex.

Substitute back

For real roots, plug x into ax² + bx + c and confirm you get (near) zero.

Frequently Asked Questions

Then the equation is linear, not quadratic. This solver requires a ≠ 0.

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Quadratic Formula

The discriminant decides the root type before any square root is taken.

Formula

D = b² − 4ac

Discriminant

D = b² − 4ac

Real roots

x = (−b ± √D) / (2a)

Complex roots

x = (−b ± √|D| i) / (2a)

Vertex

x = −b/(2a)

Scientific Background

Assumptions: real coefficients and a ≠ 0. Limitations: near-singular discriminants may need exact arithmetic; this worksheet uses floating-point.

Sources & review

Quadratic Equation Solver documents the references behind its formulas and assumptions. Always treat results as educational estimates, not personalized professional advice.

Last reviewed

July 22, 2026

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