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Quadratic Equation Solver

Solve ax² + bx + c = 0 with full working: real or complex roots, discriminant Δ, vertex (min/max), axis of symmetry, Vieta's formulas and factored form.

Enter coefficients for ax² + bx + c = 0
x2 +
x +
= 0
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How to Solve Quadratic Equations?

A quadratic equation is a second-degree polynomial equation in a single variable x with the form:

ax2 + bx + c = 0

The solutions (roots) of the equation can be found using the quadratic formula:

x = (-b ± √(b2 - 4ac)) / (2a)

The discriminant (Δ = b² - 4ac) determines the nature of the roots:

  • If Δ >0: Two distinct real roots
  • If Δ = 0: One repeated real root (two equal roots)
  • If Δ < 0: Two complex conjugate roots

Applications of Quadratic Equations

Quadratic equations describe any process where two variables are related by a degree-2 polynomial, which makes them ubiquitous in science and engineering. In physics, the height of a projectile under constant gravity is y(t) = y₀ + v₀t − ½gt², a quadratic whose positive root is the time of impact. In optics and architecture, the cross-section of every parabolic reflector — from car headlights to the 305 m Arecibo radio telescope dish — is the graph y = ax². In electrical engineering, the resonant frequency of an RLC circuit and the optimal load impedance for maximum power transfer both reduce to solving a quadratic in ω or R. In finance, break-even analysis with quadratic cost curves and yield-to-maturity calculations on bonds with two future cash flows are quadratic in the discount rate. Even Kepler's laws of planetary motion, the ballistic trajectory of every cannon shell in the 17th century, and the modern AdaBoost weight-update rule in machine learning all come back to ax² + bx + c = 0.

About this quadratic equation solver

Enter the three coefficients a, b, c of the equation ax² + bx + c = 0 (a must be non-zero) and the solver returns both roots, the discriminant Δ with a colour-coded verdict badge, and a full Root analysis panel: the vertex of the parabola, the axis of symmetry, Vieta's formulas (sum −b/a and product c/a of the roots) and the factored form a(x − x₁)(x − x₂). The complete working is also printed in the 'Solution steps' box. Real roots are shown in decimal form rounded to 10 significant digits, and complex roots in p ± qi form when Δ < 0. A true double root is detected with a floating-point tolerance, so coefficients entered as decimals or expressions still report Δ = 0 instead of two roots differing in the last digit. The calculation runs entirely in your browser — no server round-trip — so it works offline once the page is loaded.

Frequently Asked Questions

For any quadratic equation ax² + bx + c = 0 with a ≠ 0, the two roots are x = (−b ± √(b² − 4ac)) / (2a). The ± sign produces the two solutions: one with + gives x₁, one with − gives x₂. The formula is derived by completing the square on the general form and is the foundation that every other method (factoring, graphing) ultimately reduces to. The solver applies it directly and prints each substitution step.

The discriminant Δ = b² − 4ac is the part of the quadratic formula under the square root, and its sign decides what kind of roots you get. Δ > 0: two distinct real roots (the parabola crosses the x-axis twice). Δ = 0: one repeated real root (the parabola just touches the x-axis at its vertex). Δ < 0: two complex conjugate roots of the form p ± qi (the parabola never touches the x-axis). The discriminant is shown in its own field above the roots so you can read it off without inspecting the parabola.

With a = 1, b = −3, c = 2: discriminant Δ = (−3)² − 4·1·2 = 9 − 8 = 1, so two distinct real roots. Applying the formula: x = (3 ± √1) / 2 = (3 ± 1) / 2, giving x₁ = 2 and x₂ = 1. This is the default example loaded into the calculator — press Solve to see exactly the working above appear in the steps box.

Yes. When the discriminant is negative, the solver computes √(−Δ) as an imaginary part and prints the roots in the form p ± qi, where p = −b/(2a) and q = √(−Δ)/(2|a|). For example x² + x + 1 = 0 has Δ = 1 − 4 = −3, so x = (−1 ± √3·i)/2 = −0.5 ± 0.866…·i. The imaginary magnitude q is always shown as a positive value, so the two conjugates are labelled correctly even when a is negative.

No. If a = 0 the equation becomes linear (bx + c = 0), with the single solution x = −c/b (assuming b ≠ 0). The solver requires a non-zero a and will warn you otherwise. For a linear equation use a simple linear solver — the quadratic formula has 2a in its denominator, so plugging in a = 0 would divide by zero.

If x₁ and x₂ are the two roots of ax² + bx + c = 0, then the equation factors as a(x − x₁)(x − x₂) = 0. The relationships x₁ + x₂ = −b/a and x₁·x₂ = c/a (Vieta's formulas) come straight from expanding that product. So factoring and applying the quadratic formula always give the same roots; factoring is usually faster when a, b, c are small integers, while the formula always works.

The vertex sits on the axis of symmetry x = −b/(2a). Substituting that x back gives the vertex y-value: y = c − b²/(4a) = −Δ/(4a). For y = x² − 3x + 2 the vertex is at (1.5, −0.25). The vertex is the minimum of the parabola when a > 0 and the maximum when a < 0 — useful for optimization problems where you want the smallest or largest possible value of the quadratic. All of this is shown in the Root analysis panel below the roots, labelled minimum or maximum according to the sign of a.

Press Solve and read the Root analysis panel under the roots. It lists the vertex (x, y) labelled minimum (a > 0) or maximum (a < 0) — the exact deliverable for optimization problems like beam-deflection minima, max-power transfer or break-even/profit-max — plus the axis of symmetry x = −b/(2a), Vieta's sum (−b/a) and product (c/a) of the roots as quick sanity checks, and the factored form a(x − x₁)(x − x₂). For a real repeated root the factored form collapses to a(x − r)², and for complex roots it is written as a((x − p)² + q²).

Projectile motion: height as a function of time is quadratic, so finding when a thrown ball hits the ground is solving a quadratic. Geometry: the area of a rectangle whose sides differ by a fixed amount leads to a quadratic. Finance: compound-interest 'find the rate' problems and break-even pricing models reduce to quadratics. Engineering: beam deflection, parabolic reflectors, antenna design and RLC circuit resonance all hinge on quadratic relationships.
Quadratic Equation Solver — Solve ax² + bx + c = 0 with full working: real or complex roots, discriminant Δ, vertex (min/max), axis of symmetry, Vie
Quadratic Equation Solver
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Math CalculatorsMATH CALCULATORS32
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