Quadratic equations are the workhorses of algebra and applied mathematics.
About this calculator
Quadratic equations are the workhorses of algebra and applied mathematics. Anything that involves area, projectile motion, optimisation in two variables, simple economics or even trajectory of a thrown object boils down to ax² + bx + c = 0. The general solution -- the famous quadratic formula -- has been known since at least the 9th century work of al-Khwarizmi and remains one of the first non-trivial algorithms students learn.
This calculator solves any quadratic instantly and gives you not just the roots but the complete shape of the parabola: vertex, axis of symmetry, direction of opening (upward when a > 0, downward when a < 0), and the discriminant which classifies the roots. When the discriminant is negative, real roots don't exist -- the parabola never crosses the x-axis. The calculator then returns the complex conjugate pair (a ± bi) which still satisfy the equation in the broader complex number system.
Quadratic equations also model real-world things directly. The parabolic trajectory of a ball, profit-maximising price as a function of demand, the focus of a satellite dish, the path of a stream of water from a fountain -- all quadratic. The vertex represents the maximum or minimum, which is why optimisation problems in physics and economics so often reduce to finding the vertex of a quadratic. For more advanced equation solving use the [[probability]] calculator for combinatorial problems or the [[permutation-combination]] calculator for counting; pair with the [[cone]], [[cylinder]] and [[sphere]] calculators for geometry where surface areas and volumes use related polynomial relationships.
Common uses
- Solving Class 10/11/12 Mathematics homework problems involving quadratics
- Engineering problems -- projectile motion, beam deflection, AC circuit analysis
- Economics -- finding profit-maximising price or break-even quantity
- Optimisation -- finding the vertex of any parabolic function
- Computer graphics and physics simulations involving collision detection
Frequently asked questions
What does it mean when the discriminant is negative?
The parabola never crosses the x-axis, so there are no real solutions. The two roots exist in the complex number system as a ± bi pair. They are useful in physics (oscillations, AC circuits) and signal processing.
How do I factorise after finding the roots?
If r₁ and r₂ are the two roots, then a x² + b x + c factorises as a (x − r₁) (x − r₂). For example, x² − 5x + 6 = 0 has roots 2 and 3, so it factors as (x − 2)(x − 3).
Why does a need to be non-zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), with at most one solution. The calculator detects this and falls back to solving the linear case.
How is the vertex useful?
The vertex (−b/(2a), c − b²/(4a)) is the maximum point if a < 0 (parabola opens down) or minimum if a > 0. Many real-world optimisation problems -- cheapest price, longest flight, lowest cost -- come down to finding this point.
What's the relationship between sum and product of roots?
For ax² + bx + c = 0 with roots r₁ and r₂: sum = r₁ + r₂ = −b/a, product = r₁ × r₂ = c/a. These are Vieta