Quadratic Equation Solver: what this tool is for
A quadratic equation is a second-degree polynomial algebraic equation in the standard canonical form ax² + bx + c = 0, where a, b, and c are real numbers with a ≠ 0. Pioneered by Babylonian mathematicians and formally codified by Persian scholar Al-Khwarizmi in the 9th century, quadratic equations form the core bedrock of modern algebra and calculus.
The universal method for solving any quadratic equation relies on the discriminant, denoted as Δ (Delta) and defined as Δ = b² - 4ac. The algebraic sign of this discriminant dictates the exact number and nature of solutions: two distinct real roots if Δ > 0, one real double root if Δ = 0, or two complex conjugate roots in the complex number plane ℂ if Δ < 0.
Geometrically, the quadratic function f(x) = ax² + bx + c defines a parabola in the Cartesian plane with vertex coordinates S(-b/(2a), -Δ/(4a)). The solutions to the equation correspond precisely to the x-intercepts where the parabola crosses the horizontal x-axis.
This free online Quadratic Equation Solver instantly calculates real and complex roots, discriminant Delta, canonical vertex form, factorized form, parabola apex vertex, and delivers a rigorous step-by-step mathematical demonstration.
No file stored
Nothing is saved on our servers, during or after processing.
Local processing
The computation runs in your browser whenever the technology allows it.
Instant result
No waiting and no export queue: the result appears live.
Free, no account
No sign-up, no daily limit and no intrusive ads.
What this tool gives you
Complete step-by-step algebraic solution
Inspect every stage of the resolution: coefficient substitution, discriminant computation (Δ = b² - 4ac), radical simplification, and root derivation.
Full complex conjugate root support
When Δ < 0, the solver accurately computes and renders complex numbers in standard Cartesian form (x = α ± iβ).
Canonical vertex form & symmetry axis
Instantly identify parabola vertex coordinates S(α, β), vertical axis of symmetry (x = -b/2a), and canonical expression f(x) = a(x - α)² + β.
Vieta's formulas and polynomial factorization
Review the sum of roots (S = -b/a), product of roots (P = c/a), and factored polynomial product a(x - x₁)(x - x₂).
Real-world examples
Examine quadratic resolution scenarios based on the discriminant sign:
How to use Quadratic Equation Solver
Input coefficients a, b, and c
Enter your polynomial figures (e.g., a = 1, b = -5, c = 6). Coefficient 'a' must be non-zero.
Check the discriminant Delta
Evaluate Δ = b² - 4ac to determine whether the equation has real or complex solutions.
Review roots and vertex coordinates
Inspect exact and decimal values for roots x₁ and x₂, along with the parabola vertex S(α, β).
Inspect canonical and factorized forms
Copy the canonical form and polynomial factorization for homework and algebra problem sets.
Tips and mistakes to avoid
Vieta's root inspection shortcut
If the sum of coefficients a + b + c = 0, then x₁ = 1 is an immediate root and x₂ = c/a. If a - b + c = 0, then x₁ = -1 and x₂ = -c/a. For example, in 2x² - 5x + 3 = 0, since 2 - 5 + 3 = 0, the roots are effortlessly 1 and 3/2 = 1.5!
Finding the parabola vertex effortlessly
The vertex x-coordinate α is always the exact midpoint between the two real roots: α = (x₁ + x₂) / 2 = -b / (2a). Then simply evaluate f(α) to find the vertical coordinate β.
Avoid : Dropping parentheses when squaring negative b coefficients
If b = -5, squaring it yields (-5)² = +25, not -25. Forgetting that negative numbers square to positive values leads to erroneous negative discriminants.
Avoid : Dividing by 2 instead of 2a in the denominator
The denominator in the quadratic formula is 2a, not just 2. If a = 3, the denominator is 2 × 3 = 6.
Frequently asked questions
Answers to the questions people ask about Quadratic Equation Solver.
Related tools
Tools people open right after this one.
Calculate average speed, distance, or travel duration with running pace predictions and unit conversions.
Calculate kinetic energy (1/2 mv²), velocity, or mass of a moving object with drop height and impact force equivalency.
Calculate voltage (V), current (I), resistance (R), and electric power (P) with Ohm's Law and Joule's Law.
Updated on August 2026. See all tools