Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0. Real + complex roots, discriminant, vertex form, axis of symmetry, factored form. Step-by-step solution shown.
Quadratic Equation Solver
How to use the Quadratic Equation Solver
Identify a, b, c from your equation
Standard form: ax² + bx + c = 0. Coefficient of x² is a; coefficient of x is b; constant is c. If your equation has terms on both sides, move everything to one side first to get the standard form. Coefficients can be positive, negative, or fractional — the tool handles all real numbers.
Read x₁ and x₂
The two roots (solutions). If the discriminant (b²−4ac) is positive, you get two distinct real roots. If zero, one repeated real root (the parabola touches the x-axis at exactly one point). If negative, two complex conjugate roots — the parabola doesn\'t cross the x-axis.
Check vertex + axis of symmetry
Vertex (h, k) is the parabola\'s extreme point: minimum if a > 0 (opens upward), maximum if a < 0 (opens downward). h = −b/(2a) is also the axis of symmetry. Useful for: optimisation problems, projectile motion (max height = vertex k), and graphing.
Use the factored form
For real roots, the equation factors as a(x − r₁)(x − r₂) = 0. The factored form is useful for: simplifying related expressions, checking integer-coefficient equations, and graphing. Complex roots can\'t be factored in real numbers (would require imaginary coefficients).
The quadratic equation — one of math's most-used tools
Quadratic equations (degree-2 polynomial equations of form ax² + bx + c = 0) are everywhere — projectile motion, optimisation, finance, engineering, computer graphics. The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, has been known in some form since Babylonian times (c. 2000 BCE). The modern formula was published by Indian mathematician Brahmagupta around 628 CE. It\'s typically the first formula students memorise that handles BOTH real + complex solutions automatically — a small but profound demonstration that math sometimes requires extending the number system to give answers to natural questions.
The discriminant — predicting the nature of solutions
Before computing the full formula, the discriminant Δ = b² − 4ac tells you what kind of solutions you\'ll get. Δ > 0: two distinct real solutions; parabola crosses x-axis at two points; equation has two different x-values that satisfy it. Δ = 0: one repeated real solution; parabola touches x-axis at exactly one point (the vertex); both formulaic solutions equal −b/2a. Δ < 0: two complex conjugate solutions (a ± bi); parabola doesn\'t touch x-axis; only solutions exist in the complex plane. Practical implication: in physics + engineering, Δ < 0 often signals an impossible scenario (negative time, imaginary velocity). In pure math, complex solutions are first-class citizens.
The quadratic formula handles both real + complex solutions automatically — one of the first demonstrations in math curriculum that extending the number system gives useful answers to natural questions.
Vertex form + completing the square
The vertex form a(x − h)² + k rewrites the quadratic to expose its geometric properties directly. (h, k) is the vertex; x = h is the axis of symmetry. For a > 0 (opens upward), k is the minimum value; for a < 0 (opens downward), k is the maximum. Conversion: h = −b/(2a); k = c − b²/(4a). The derivation uses "completing the square" — the same technique used to derive the quadratic formula itself. Use cases: optimisation problems (max revenue at vertex), projectile peak altitude, parabolic reflector focal point, beam stress analysis.
Common quadratic applications
Physics — projectile motion: position vs time follows h(t) = −½gt² + v₀t + h₀. Solving h(t) = 0 gives time of impact; vertex gives peak altitude + time. Finance — break-even analysis: revenue function minus cost function often produces a quadratic; roots are break-even points. Engineering — beam deflection: simple beam under uniform load has quadratic deflection profile. Computer graphics — Bezier curves: quadratic Bezier curves use the form ax² + bx + c; evaluation involves solving quadratics. Optimisation: many optimisation problems reduce to finding vertex of quadratic objective functions (especially in portfolio theory, machine learning loss surfaces). Knowing the quadratic formula + vertex form is foundational across all of these.
10 Things to Know About Quadratic Equations
Standard form: ax² + bx + c = 0 where a ≠ 0 (else it\'s linear).
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Solves any quadratic in real or complex numbers.
Discriminant Δ = b² − 4ac. Predicts solution type: positive = 2 real distinct; zero = 1 repeated; negative = 2 complex.
Vertex (h, k) = (−b/2a, c − b²/4a). Parabola\'s extreme point.
For a > 0, parabola opens upward (vertex = minimum). For a < 0, opens downward (vertex = maximum).
Axis of symmetry: x = −b/(2a). Vertical line through the vertex.
Quadratic formula known since ~2000 BCE (Babylonian mathematicians). Modern form by Brahmagupta, India, c. 628 CE.
The formula\'s derivation uses "completing the square" — converting standard form to vertex form algebraically.
Common applications: projectile motion, break-even analysis, optimisation, Bezier curves, beam deflection.
Quadratics with negative discriminant give complex conjugate roots — the first formal use of imaginary numbers in school curriculum.
Frequently Asked Questions
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Then it\'s not a quadratic equation — it\'s linear (degree 1). The tool handles this gracefully: with a=0, b≠0, c: x = −c/b (linear solution). With a=0, b=0, c=0: infinite solutions (trivially true). With a=0, b=0, c≠0: no solution (impossible equation). Quadratic formula requires a ≠ 0.
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When the discriminant (b² − 4ac) is negative. This happens when the parabola doesn\'t cross the x-axis. In physics: often signals an impossible scenario in the real world (negative time, imaginary velocity, etc.) — the equation has no physical meaning. In pure math: complex solutions are first-class objects in algebra, signal processing, quantum mechanics. For school + most engineering: "no real solutions" is the practical answer; complex roots are notation.
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Vertex form
a(x − h)² + kexposes the parabola\'s geometric properties directly. (h, k) = vertex coordinates. For optimisation: k is the extreme value (max or min depending on a\'s sign). For graphing: h tells you horizontal shift; k tells you vertical shift. For physics: in projectile motion, h is time of peak; k is peak altitude — directly readable from vertex form. -
Many quadratics have irrational or complex roots that don\'t simplify into clean integer factors. "Nice" factoring works when roots are rational + small. Example: x² − 5x + 6 = (x − 2)(x − 3) — clean. x² − 5x + 7: roots ≈ 2.382 and 2.618 — doesn\'t factor over integers. x² + 1: roots ±i — doesn\'t factor over real numbers. The quadratic formula always gives an answer; factoring is a shortcut for specific clean cases.
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Three-step process. (1) Set up the equation: identify the variable, write the relationship, get to standard form ax² + bx + c = 0. (2) Solve using this tool or the formula. (3) Interpret: not all mathematical solutions are physically meaningful. Negative time, imaginary velocity, lengths < 0 should be discarded. Most physics problems give two roots; pick the one that makes physical sense.
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It\'s derived from "completing the square." Starting with ax² + bx + c = 0: (1) divide by a → x² + (b/a)x + (c/a) = 0; (2) move c/a to other side → x² + (b/a)x = −c/a; (3) add (b/2a)² to both sides → x² + (b/a)x + (b/2a)² = (b/2a)² − c/a; (4) factor left as (x + b/2a)² = b²/4a² − c/a = (b² − 4ac)/4a²; (5) take square root → x + b/2a = ±√(b² − 4ac)/(2a); (6) solve for x → x = (−b ± √(b² − 4ac))/2a. The formula is just a compact summary of this 6-step algebraic process.
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An algebraic technique that rewrites ax² + bx + c as a perfect-square term plus a constant: a(x + h)² + k. Used to: (1) derive the quadratic formula; (2) convert standard form to vertex form; (3) solve quadratics by hand without the formula; (4) prove identities. Modern students typically learn the formula directly; completing the square is taught as derivation + appears in physics/engineering problems involving Gaussian distributions, oscillators, etc.
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This tool handles only quadratics (degree 2). Cubic equations (degree 3, ax³ + bx² + cx + d = 0) have a closed-form solution (Cardano\'s formula, 1545) but it\'s much more complex; usually solved numerically. Quartic equations (degree 4) also have a closed-form solution (Ferrari\'s formula). Degree 5+: provably no closed-form solution via radicals (Abel-Ruffini theorem, 1824). For higher-degree equations, use numerical methods (Newton-Raphson) or computer algebra systems (Wolfram Alpha, SymPy, MATLAB).
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No. All calculations run in your browser via JavaScript. Open DevTools → Network and confirm zero outbound requests. Coefficients stay on your device. Safe for homework, research, or any private calculations.
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Pair with: Triangle Solver (RT-CNV-095) for trigonometry; Ohm\'s Law Calculator (RT-CNV-096) for electrical math; Linear Regression (RT-CNV-084) for statistics. External: Wolfram Alpha for general math problems; Desmos for graphing; Khan Academy for tutorials; GeoGebra for visual algebra exploration; SymPy (Python) or Mathematica for symbolic computation.
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