📐 Quadratic Equation Formula Solver

A quadratic equation solver for ax² + bx + c = 0: the discriminant step by step, real or complex roots, and the vertex of the parabola.

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Quadratic Standard Form: ax² + bx + c = 0

Coefficient a (≠ 0)

Coefficient b

Constant c

Two Distinct Real Roots

x₁ = 3.0000

x₂ = 2.0000

Discriminant (Δ = b² - 4ac)

Δ = 1.00

Parabola Vertex (h, k)

(2.50, -0.25)

What Quadratic Equation Formula Solver Does

The quadratic formula solves every equation of the form ax² + bx + c = 0, and the part that carries the meaning is the piece under the square root. The discriminant, b² − 4ac, tells you what kind of answer to expect before you have finished computing it: positive means two real roots, zero means one repeated root, negative means the parabola never crosses the x-axis and the roots are complex.

That is worth knowing because a negative discriminant is not an error. It is a complete answer — the curve sits entirely above or entirely below the axis — and a solver that reports "no solution" there is being imprecise. There are two solutions; they are just not real numbers.

The vertex is the other thing worth having. It sits at x = −b ÷ 2a, exactly halfway between the roots when they are real, and it is the maximum or minimum of the curve depending on the sign of a. For anything modeled with a quadratic — projectile height, profit against price — the vertex is usually the answer you actually wanted.

This shows all of it: the discriminant with its interpretation, both roots real or complex, the vertex, and which way the parabola opens.

How to Use Quadratic Equation Formula Solver

Enter coefficients a, b, and constant c (where a ≠ 0)
Review discriminant calculation and root classification
Inspect root values x₁ and x₂ and vertex coordinates (h, k)

Formula Used by Quadratic Equation Formula Solver

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a, with vertex at x = −b ÷ 2a

b² − 4ac: the discriminant — its sign determines the nature of the roots
a ≠ 0: if a is zero the equation is linear, not quadratic, and the formula divides by zero

Worked example

x² − 5x + 6 = 0

Discriminant: (−5)² − 4×1×6 = 25 − 24 = 1
Positive, so two distinct real roots
x = (5 ± 1) ÷ 2 → 3 and 2
Vertex: x = 5 ÷ 2 = 2.5, y = 6 − 25÷4 = −0.25

Result: Roots at 3 and 2, vertex at (2.5, −0.25) — the minimum, since a is positive so the parabola opens upward.

What the discriminant tells you

b² − 4ac	Roots	The graph
Positive	Two distinct real roots	Crosses the x-axis twice
Zero	One repeated real root	Touches the x-axis at the vertex
Negative	Two complex conjugate roots	Never touches the x-axis

How to Read Your Result

A negative discriminant is an answer, not a failure

The roots exist as complex conjugates — a pair of the form p ± qi. In a physical model that usually means the situation you described never occurs: a projectile that never reaches the height you asked about, for instance. The mathematics is telling you something rather than breaking.

The vertex is often the real question

Maximum profit, peak height, minimum cost — all of them are the vertex rather than a root. If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.

Check that a is not zero

With a of zero there is no x² term and the equation is linear — bx + c = 0, with the single solution −c ÷ b. The quadratic formula divides by 2a and fails. It is the most common way this calculation goes wrong when the coefficients come from somewhere else.

Limitations & Accuracy Notes

Real coefficients only; complex inputs are not accepted, though complex roots are produced and displayed.
Floating-point arithmetic, so results are shown to a fixed number of decimals rather than as exact surds — √2 appears as 1.4142, not as a radical.
Very large or very small coefficients can lose precision, and subtracting nearly equal numbers in the formula is a known source of that.
It does not factorize, complete the square, or show a step-by-step derivation.
Higher-degree polynomials are outside its scope entirely.

Frequently Asked Questions

How does the quadratic discriminant determine root types?

If Δ > 0, there are two distinct real roots. If Δ = 0, there is one real double root. If Δ < 0, there are two complex conjugate roots with imaginary unit i.

What is the standard quadratic formula?

x = (-b ± √(b² - 4ac)) / (2a).

What does the discriminant tell me?

b² − 4ac decides the nature of the roots before you finish solving. Positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots. Checking it first tells you what kind of answer to expect.

What if a equals zero?

Then it is not a quadratic — it collapses to the linear equation bx + c = 0, with a single root of −c/b. A solver that divides by a without checking produces an infinity or a NaN rather than the correct linear answer.

Why do I get complex roots?

Because the parabola never crosses the x-axis. The roots still exist and still describe the equation, they simply are not real numbers — which is a perfectly valid outcome rather than an error, and matters in engineering contexts such as oscillation and AC circuits.

Should I factorize or use the formula?

Factorizing is quicker when the roots are simple integers and it is worth spotting. The formula always works, which makes it the reliable default — and many exam schemes award marks for either as long as the working is shown.

Does it show the working?

The page states the formula, defines each term and works through a numeric example, so a result can be checked by hand rather than taken on trust.

Is my data stored?

No. The calculation runs in your browser.

References & Further Reading

Wolfram MathWorld — Quadratic Formula — Derivation, the discriminant, and the numerical-stability caveat

By OnlineToolHubs Team • September 2026