Quadratic Formula Calculator

The discriminant decides everything before you solve.

The discriminant decides everything before you solve. Roots of ax² + bx + c with the discriminant, the vertex and the axis of symmetry.

Written and maintained by Mohit PatelLast checked August 4, 2026How we build these

The coefficient of x²

Two real roots

x = 3, 2

Discriminant 1 · vertex at (2.5, -0.25)

Discriminant b² − 4ac1

Nature of the rootstwo real roots

Root 13

Root 22

Sum of roots (−b/a)5

Product of roots (c/a)6

Vertex(2.5, -0.25)

Axis of symmetryx = 2.5

Parabola opensUpward

The discriminant decides everything before the roots are calculated: positive gives two real roots, zero gives one repeated root at the vertex, negative gives a complex pair. Vieta's relations are the useful check — the roots should sum to −b/a and multiply to c/a.

How the Quadratic Formula Calculator works

Roots of ax² + bx + c with the discriminant, the vertex and the axis of symmetry. Complex roots are reported rather than refused, because no real solution is only true over the reals — the parabola is still a real curve.

Also known as: solve a quadratic equation · discriminant calculator · find the roots of ax2+bx+c · parabola vertex calculator

The discriminant answers the question before you solve it

b² − 4ac is the quantity under the root, and its sign decides everything. Positive gives two real roots, zero gives one repeated root exactly at the vertex, negative gives a complex conjugate pair.

That is useful before doing any further work. Knowing a parabola never crosses the axis, or touches it once, frequently answers the underlying question without the roots being needed at all.

It also explains the geometry. A zero discriminant means the vertex sits on the x-axis; a negative one means the whole parabola is on one side of it. The algebra and the picture describe the same fact.

Where the formula comes from

It is completing the square, done once on the general form. Divide through by a, move the constant across, add the square of half the x-coefficient to both sides, take the root and rearrange — the formula falls out.

Which means completing the square and using the formula are the same method at different levels of abstraction. Doing it by hand on a specific equation is often quicker than substituting into the formula, and it always shows the vertex.

Vieta's relations come from the same place: the roots sum to −b/a and multiply to c/a. They are the fastest possible check on a solution, and they hold for complex roots too.

Complex roots are not nothing

A negative discriminant is frequently reported as no solution, which is only true over the reals. The roots exist in the complex numbers and they carry real physical meaning in several fields.

In engineering and physics they describe oscillation. The real part gives growth or decay and the imaginary part gives a frequency, which is central to control theory, signal processing and any system that rings.

The parabola itself remains entirely real regardless. Its vertex, axis of symmetry and shape are unaffected by whether it crosses the axis, which is why those figures are reported even when the roots are complex.

Where to go next

The Quadratic Formula question rarely arrives on its own. These are the ones that usually come with it:

Slope Intercept Calculator — A vertical line has undefined slope, not infinite.
Nth Root Calculator — Every root is a fractional power.
Polygon Area Calculator — Exterior angles always sum to 360°, whatever the shape.
Percentage Calculator — Every common percentage question in one place.

Frequently asked questions

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any quadratic equation, and it comes from completing the square on the general form.

What does the discriminant tell me?

b² − 4ac decides the nature of the roots before you calculate them. Positive gives two real roots, zero gives one repeated root at the vertex, negative gives a complex pair.

What if the discriminant is negative?

The parabola never crosses the x-axis, so there are no real roots. The complex roots are still meaningful, and the vertex and shape of the curve are perfectly real, which is why they are shown.

How do I find the vertex?

The x-coordinate is −b ÷ 2a, which is also the axis of symmetry. Substitute it back to get the y-coordinate. The vertex is the minimum if a is positive and the maximum if negative.

What are Vieta's formulas?

The roots sum to −b/a and multiply to c/a. They are the fastest check on a solution, and they work even when the roots are complex.

Can I solve a quadratic by factoring instead?

When the roots are rational, yes, and it is quicker. The formula always works, which is why it is the fallback when factoring does not present itself.

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