Calculadora de la fórmula cuadrática
Resuelve cualquier ecuación de segundo grado ax² + bx + c = 0 con raíces reales o complejas, y obtén el discriminante, el vértice, el eje de simetría y la forma factorizada.
Cómo usar esta calculadora
- 1Write your equation in the standard form ax² + bx + c = 0, moving every term to one side.
- 2Enter a, b, and c — including their signs. A missing x term means b = 0.
- 3Read the roots, and check the discriminant to understand why you got real or complex answers.
Cómo funciona
La fórmula cuadrática
x = (−b ± √(b² − 4ac)) ÷ 2a Discriminante: Δ = b² − 4ac Vértice: x = −b ÷ 2a, y = f(−b ÷ 2a) Suma de raíces = −b ÷ a, Producto de raíces = c ÷ a
La fórmula cuadrática resuelve cualquier ecuación de la forma ax² + bx + c = 0. El discriminante, la parte bajo la raíz (b² − 4ac), decide el tipo de solución: si es positivo hay dos raíces reales distintas, si es cero una raíz doble, y si es negativo dos raíces complejas conjugadas. El vértice y el eje de simetría describen la parábola asociada.
Ejemplo resuelto
Para x² − 3x + 2 = 0: el discriminante es (−3)² − 4·1·2 = 1, positivo, así que hay dos raíces reales. La fórmula da x = (3 ± 1) ÷ 2, es decir x = 2 y x = 1, y la forma factorizada es (x − 1)(x − 2).
Calculadora de la fórmula cuadrática: la guía completa
What the discriminant tells you before you solve
The discriminant b² − 4ac is worth computing first, because it answers the shape of the problem in one number. Positive means the parabola crosses the x-axis twice and there are two distinct real solutions. Zero means it touches the axis at exactly one point — a repeated root, and the vertex sits on the axis. Negative means the parabola never reaches the axis, so there are no real solutions and the roots are a complex conjugate pair.
A discriminant that is a perfect square carries extra information: it means the quadratic factors cleanly over the rationals, so factoring by inspection will work and you did not need the formula at all.
Three ways to solve, and when each is fastest
Factoring is fastest when it works — if you can spot two numbers that multiply to ac and add to b, you are done in seconds. It only works when the roots are rational.
Completing the square is slower but it is where the quadratic formula comes from, and it is the method that converts a quadratic into vertex form a(x − h)² + k, which is what you want for graphing or optimisation.
The quadratic formula always works. It handles irrational roots, complex roots, and ugly decimal coefficients identically, which is why it is the method to reach for when the numbers are not friendly.
Reading the vertex and the parabola
The vertex is the turning point. Its x-coordinate is always −b ÷ 2a, sitting exactly halfway between the two roots because a parabola is symmetric about that line. Substituting that x back into the equation gives the y-coordinate.
The sign of a decides which way the curve opens: positive opens upward so the vertex is the minimum, negative opens downward so the vertex is the maximum. This is the whole basis of quadratic optimisation problems — maximum profit, maximum projectile height, minimum material cost all reduce to finding a vertex.
Vieta's formulas as a check
For any quadratic, the two roots must sum to −b ÷ a and multiply to c ÷ a. These are Vieta's formulas, and they give a free five-second check on any answer.
For x² − 5x + 6 = 0, the roots 3 and 2 sum to 5 (which is −(−5) ÷ 1) and multiply to 6 (which is 6 ÷ 1). If your roots fail either check, you have made an arithmetic slip. The check also works for complex roots, where the imaginary parts cancel in the sum and combine in the product.
Preguntas frecuentes
¿Qué indica el discriminante?
El signo de b² − 4ac indica el tipo de raíces: positivo, dos raíces reales distintas; cero, una raíz real doble; negativo, dos raíces complejas conjugadas sin cortes con el eje x. Su valor también mide cuán separadas están las raíces.
¿Qué pasa si a es cero?
Entonces la ecuación no es cuadrática sino lineal (bx + c = 0) y tiene una sola solución, x = −c ÷ b. La fórmula cuadrática requiere que a sea distinto de cero, porque de lo contrario habría una división por cero.
¿Puede una ecuación cuadrática no tener solución real?
Sí. Cuando el discriminante es negativo, la parábola no corta el eje x y no hay raíces reales; las dos soluciones son números complejos conjugados. La calculadora las muestra en su forma a ± bi.