Two distinct real roots (the discriminant is positive).
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How This Calculator Works
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x, using only the
equation’s three coefficients. Enter a, b, and c, and this calculator finds every value of
x that makes the equation true — including the case where the answer involves complex numbers
rather than real ones.
A quadratic equation always has exactly two solutions when counted correctly: two distinct real
numbers, one real number counted twice (a “repeated root”), or a pair of complex numbers. Which
case applies depends entirely on the sign of the discriminant (the part under the square
root) — this calculator tells you which one you’ve got, not just the raw numbers.
The Formula
For ax2+bx+c=0 (with a=0):
x=2a−b±b2−4ac
The term under the square root, b2−4ac, is called the discriminant:
Discriminant
What it means
Positive
Two distinct real roots
Exactly zero
One repeated real root
Negative
Two complex-conjugate roots (no real solution)
Worked Example
For x2−3x+2=0 (a=1, b=−3, c=2):
Discriminant: (−3)2−4(1)(2)=9−8=1, which is positive, so there are two distinct
real roots.
x=23±1=23±1.
x=24=2 or x=22=1.
Checking: (1)2−3(1)+2=1−3+2=0 and (2)2−3(2)+2=4−6+2=0 — both
solutions check out.
Cómo funciona esta calculadora
La fórmula cuadrática resuelve cualquier ecuación de la forma ax² + bx + c = 0 para x, usando
únicamente los tres coeficientes de la ecuación. Ingresa a, b y c, y esta calculadora
encuentra todos los valores de x que hacen verdadera la ecuación — incluyendo el caso en el que la
respuesta involucra números complejos en lugar de reales.
Una ecuación cuadrática siempre tiene exactamente dos soluciones cuando se cuentan correctamente:
dos números reales distintos, un número real contado dos veces (una “raíz repetida”), o un par de
números complejos. El caso que aplica depende por completo del signo del discriminante (la
parte bajo la raíz cuadrada) — esta calculadora te indica cuál de ellos tienes, no solo los
números en bruto.
La fórmula
Para ax2+bx+c=0 (con a=0):
x=2a−b±b2−4ac
El término bajo la raíz cuadrada, b2−4ac, se llama el discriminante:
Discriminante
Qué significa
Positivo
Dos raíces reales distintas
Exactamente cero
Una raíz real repetida
Negativo
Dos raíces complejas conjugadas (sin solución real)
Ejemplo resuelto
Para x2−3x+2=0 (a=1, b=−3, c=2):
Discriminante: (−3)2−4(1)(2)=9−8=1, que es positivo, así que hay dos raíces reales
distintas.
x=23±1=23±1.
x=24=2 o x=22=1.
Comprobación: (1)2−3(1)+2=1−3+2=0 y (2)2−3(2)+2=4−6+2=0 — ambas
soluciones se cumplen.
For an equation in the form ax² + bx + c = 0, the quadratic formula gives x = (-b ± √(b² - 4ac)) ÷ (2a). It solves for every value of x that makes the equation true, using only the equation's three coefficients.
What does the discriminant tell me?
The discriminant is the part under the square root, b² - 4ac. A positive discriminant means two distinct real roots, zero means exactly one repeated real root, and a negative discriminant means the two roots are complex (involving the imaginary unit i) rather than real numbers.
What if "a" is zero?
If a = 0, the equation isn't actually quadratic anymore — it's linear (bx + c = 0), which the quadratic formula can't solve since it would divide by zero. This calculator flags that case and tells you to use a linear equation instead.
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