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Quadratic Equation Solver

Enter the coefficients of ax²+bx+c=0 to instantly get the discriminant (Δ), real or complex roots, Vieta's formula verification, the vertex, and a parabola graph — with a full step-by-step quadratic-formula walkthrough.

Δ = b²−4ac Real & complex roots Vieta verification Updated: Jul 19, 2026
Automatically convert a free-form equation into a, b, c (e.g. 2x² + 3x = 2, x² = 5x − 6). Use ² or ^2 for x².
a must not be 0. Use a period for decimals, "−" for negatives, and "3/4" for fractions. Examples:
Enter complex roots (p ± qi)Off = enter two real roots
Build an equation from known roots using Vieta's formulas in reverse. Examples:
⚙️ Advanced settings — decimal places, step-by-step, graph
Step-by-step solutionΔ, square root, and root calculation on separate lines
Parabola graphVisual summary via inline SVG
Δ = b² − 4acDiscriminant formula
x=(−b±√Δ)/2aQuadratic formula
x₀ = −b/2aVertex (axis of symmetry)
x1+x2=−b/aVieta: sum/product of roots
Quick answer A quadratic equation has the form ax²+bx+c=0 and is solved with the quadratic formula x = (−b ± √(b²−4ac)) / 2a. First calculate the discriminant Δ=b²−4ac: if Δ>0 there are two distinct real roots, if Δ=0 there is one repeated root, and if Δ<0 there are two complex roots in the form a+bi. The calculator above applies these steps automatically.
⚙️ Accuracy & scope note: This tool solves quadratic equations with real coefficients (a, b, c ∈ ℝ) only, in the form ax²+bx+c=0; entering a=0 reduces the equation to a linear one and a separate note is shown. The formulas are based on standard algebra and analytic geometry references and are evergreen mathematical rules that do not change over time or require annual updates. Results are rounded to the decimal precision you select.
📐 Quadratic Equation Solver — ToolPico ↗

What is a quadratic equation, and how do you solve one?

A complete guide — with formulas and examples — to the discriminant, the quadratic formula, Vieta's formulas, the vertex, factoring, and completing the square.

A quadratic equation is written in the form ax²+bx+c=0 (a≠0), where the highest power of x is 2. Depending on the sign of the coefficients, this type of equation resolves to two distinct real roots, a single repeated (double) real root, or two complex roots. This tool works in both directions: finding the roots of a given equation, and building an equation from known roots (Vieta's formulas in reverse).

What is the discriminant (Δ), and how do you interpret it?

Quick answerThe discriminant Δ=b²−4ac is the term under the square root in the quadratic formula, and it determines the root structure: Δ>0 means two distinct real roots, Δ=0 means one repeated (double) real root, and Δ<0 means two complex (conjugate) roots. It's the first — and most important — step calculated before solving the equation.

What happens if the discriminant is negative? (complex roots)

Quick answerWhen Δ<0, the equation has no real roots; the roots are a pair of complex conjugates written as x=p+qi and x=p−qi (p=−b/2a, q=√|Δ|/2a). Graphically, this means the parabola never crosses the x-axis — if a>0 the parabola lies entirely above it, and if a<0 it lies entirely below it.

What is the quadratic formula (square-root formula)?

Quick answerThe quadratic formula is x = (−b ± √(b²−4ac)) / 2a. Substitute the coefficients a, b, c, first calculate Δ=b²−4ac, take its square root, and the ± sign produces two roots (or one, if Δ=0). This formula holds for every quadratic equation with real coefficients where a≠0.

What are Vieta's formulas?

Quick answerVieta's formulas describe the relationship between the roots and coefficients: the sum of the roots is x1+x2=−b/a, and the product of the roots is x1·x2=c/a. They're used to quickly verify roots you've found, to compute the sum/product without solving the equation, and to build an equation from its roots (the "Equation from Roots" mode of this tool).

How do you find the vertex of a parabola?

Quick answerFor the parabola y=ax²+bx+c, the vertex is found with x₀=−b/2a and y₀=c−b²/4a (equivalently −Δ/4a). If a>0 the vertex is a minimum, and if a<0 it's a maximum; the axis of symmetry is the vertical line x=x₀.

Factoring and completing the square

Quick answerFactoring over the real numbers is only possible when Δ≥0: a(x−x1)(x−x2). Completing the square works in every case, and rewrites the expression as a(x−x₀)²+y₀; this form shows the vertex directly and is the method the quadratic formula is derived from.

Related calculators: vertex, linear equation & Vieta

The most common related calculations for quadratic equations — all instant.

📍Vertex calculator
Quickly find the vertex and opening direction of the parabola y=ax²+bx+c.
📏Linear equation solver
Solve a first-degree equation ax+b=0 (a≠0).
🔗Vieta verifier
Generate the sum, product, and (for a=1) the b, c coefficients from two roots.

Reference tables & examples

A citable discriminant/root table, example equations, and a summary of Vieta's formulas.

Root type by sign of the discriminant (Δ)
Δ conditionNumber/type of rootsGraph / x-axis
Δ > 02 distinct real rootsCrosses the x-axis at 2 points
Δ = 01 (double) real rootTouches the x-axis at the vertex
Δ < 02 complex (conjugate) rootsNever crosses the x-axis
Example equations and their roots
EquationΔRoots
x² − 5x + 6 = 01x1=3, x2=2
x² − 4x + 4 = 00x=2 (double root)
x² + 2x + 5 = 0−16x=−1±2i
2x² + 3x − 2 = 025x1=0.5, x2=−2
x² − 1 = 04x1=1, x2=−1

Try any of these values with one click using the "Examples" chips on the calculator above.

Vieta's formulas summary (ax²+bx+c=0)
QuantityFormula
Sum of rootsx1 + x2 = −b/a
Product of rootsx1 · x2 = c/a
Vertex (x₀)−b/2a
Vertex (y₀)c − b²/4a = −Δ/4a

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Embed the quadratic equation solver on your own website for free. Copy the code below into your HTML — the tool runs in a simplified view and links back to this page as its source.

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Quadratic equation terms glossary

Short definitions of the core terms used when solving quadratic equations.

Discriminant (Δ)The expression b²−4ac; determines the number and type of roots.
RootA value of x that satisfies the equation, i.e. makes it equal zero.
Real rootA root belonging to the real numbers; exists when Δ≥0.
Complex rootA conjugate pair of roots in the form a+bi that arises when Δ<0.
ParabolaThe graph of y=ax²+bx+c; opens upward if a>0, downward if a<0.
VertexThe minimum or maximum point of the parabola: (−b/2a, −Δ/4a).
Axis of symmetryThe vertical line through the vertex, x=−b/2a, that splits the parabola in half.
Vieta's formulasThe identities x1+x2=−b/a and x1·x2=c/a linking the roots to the coefficients.
FactoringWriting the expression as a(x−x1)(x−x2); possible over the reals when Δ≥0.
Completing the squareRewriting ax²+bx+c as a(x−x₀)²+y₀.
CoefficientThe constants a, b, c in the equation; a scales x², b scales x, and c is the constant term.
y-interceptThe point (0, c) where the graph crosses the y-axis, found by setting x=0.

In-depth guides

Detailed answers to the questions people ask most about quadratic equations.

How does the discriminant tell you how many roots there are?

To understand how many roots — and of what type — a quadratic equation has, all you need to check is the sign of Δ=b²−4ac. If Δ is positive there are two distinct real roots, if it's zero there is one (double) real root, and if it's negative there are two complex roots — this single step shows you the "shape" of the equation before you finish solving it.

Enter the coefficients in the "Find Roots" tab above and click Solve — the tool calculates Δ automatically and explains which root type it corresponds to, step by step.

What are complex roots used for in real life?

The complex roots that appear when Δ<0 may look abstract, but they're used widely in electrical engineering for analyzing AC circuits, in control systems for stability analysis, and in signal processing. Mathematically, they guarantee that an equation with no real roots still always has two conjugate complex roots.

This tool displays the roots directly in a+bi form when Δ<0, and its graph summary explains why the parabola never touches the x-axis.

How do you build an equation from its roots? (reverse Vieta)

When you know two roots, you can build the equation by using Vieta's formulas in reverse: calculate b = −a(x1+x2) and c = a·x1·x2, then write ax²+bx+c=0. Choosing a=1 gives the simplest form.

Enter two real roots (or p and q for a complex root) in the "Equation from Roots" tab; the tool calculates b and c, builds the equation, and re-solves it to verify the roots.

Frequently asked questions

How do you solve a quadratic equation?
First rewrite the equation in the form ax²+bx+c=0. Calculate the discriminant Δ=b²−4ac and apply the quadratic formula x=(−b±√Δ)/2a. If Δ>0 there are two distinct real roots, if Δ=0 there is one repeated (double) real root, and if Δ<0 there are two complex roots in the form a+bi. You can verify the roots using Vieta's formulas (x1+x2=−b/a, x1·x2=c/a).
What is the discriminant?
The discriminant is the expression Δ=b²−4ac that determines the root structure of a quadratic equation. It is the term under the square root in the quadratic formula, and its sign tells you directly how many real roots exist and whether the root is single, double, or complex — it's the first thing you calculate before solving the equation.
What happens if the discriminant is negative?
When Δ<0, the equation has no real roots; the roots are a pair of complex conjugates written as x=a+bi and x=a−bi. Graphically, this means the parabola never crosses the x-axis: if a>0 the parabola lies entirely above the x-axis, and if a<0 it lies entirely below it.
What is the quadratic formula?
The quadratic formula (square-root formula) is x=(−b±√(b²−4ac))/(2a). Substitute the coefficients a, b, c, first calculate the discriminant b²−4ac, take its square root, and the ± sign gives you two roots (or one, if Δ=0). When a=0, the equation is no longer quadratic.
What are Vieta's formulas?
Vieta's formulas relate the roots of ax²+bx+c=0 to its coefficients: the sum of the roots is x1+x2=−b/a, the product of the roots is x1·x2=c/a. They're useful for finding the sum/product of the roots without solving the equation, for quickly verifying roots you've already found, and for building an equation from known roots.
How do you find the vertex of a parabola?
For the parabola y=ax²+bx+c, the x-coordinate of the vertex is h=−b/(2a) and the y-coordinate is k=c−b²/(4a) (equivalently k=−Δ/4a). If a>0 the vertex is the parabola's minimum point, and if a<0 it's the maximum point; the axis of symmetry is the vertical line x=h.
When is factoring possible?
A quadratic expression can be factored over the real numbers as a(x−x1)(x−x2) only when the discriminant Δ≥0, where x1 and x2 are the real roots. When Δ=0, the expression becomes a perfect square, a(x−x1)². When Δ<0, it cannot be factored over the reals — only over the complex numbers.
What is completing the square?
Completing the square is a method that rewrites ax²+bx+c as a(x−h)²+k, where h=−b/(2a) and k=c−b²/(4a) are the vertex coordinates. This form lets you solve the equation without deriving the quadratic formula, and lets you read the parabola's vertex directly.
How are the roots shown in exact (radical/fraction) form?
The exact result comes from simplifying the square root inside x=(−b±√Δ)/2a: if the discriminant Δ isn't a perfect square, √Δ is reduced to its simplest radical form g√m (e.g. x=(3±√17)/2); if it is a perfect square, the roots are written as rational fractions or integers (e.g. x1=1/2, x2=−2). This calculator shows this exact (radical/fraction) form alongside the decimal rounding — it's the precise result expected on tests and homework.
Can I solve an equation without first converting it to ax²+bx+c=0?
Yes. Type a free-form equation (e.g. x²=5x−6 or 2x²+3x=2) into the "Paste an equation" field and click Parse — the tool combines both sides, simplifies, and reduces the equation to a, b, c coefficients, then solves it automatically. You can type ² or ^2 for x², and fractional coefficients like 3/4 are supported.
How does Practice Mode work?
Practice Mode (a separate tab on the calculator) generates a random quadratic equation at your chosen difficulty — Easy (integer roots, a=1), Medium (integer roots, a≠1), or Hard (may include fractional or irrational roots) — and lets you type in the root(s) to check your work instantly against the same solver engine used elsewhere on the page. Click "New random equation" for another problem at any time; your running score is shown above the answer fields.

Methodology & sources

ToolPico's Quadratic Equation Solver is a free, independent tool. The calculation engine applies standard algebra directly: the discriminant Δ=b²−4ac determines the root type; real roots use the quadratic formula x=(−b±√Δ)/2a, complex roots use x=−b/2a ± (√|Δ|/2a)i; the vertex is found by completing the square, and roots are verified with Vieta's formulas (x1+x2=−b/a, x1·x2=c/a). Results are computed instantly client-side (in your browser); no data is sent to a server.

Basis: Classical algebra and analytic geometry (the discriminant/square-root method, Vieta's theorems, completing the square). These rules are evergreen mathematical facts that do not require annual updates. Last updated: July 19, 2026.
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