Name: complex roots of quadratic equations
Uploaded: 2025-11-26T05:44:28.000Z
Duration: 1 min 29 s
Channel: JoVE

Las ecuaciones cuadráticas en las que el discriminante —la expresión bajo la raíz cuadrada en la fórmula cuadrática— tiene un valor negativo no tienen soluciones reales.Para resolverlos, el sistema numérico se amplía para incluir números complejos utilizando la unidad imaginaria i, definida como la raíz cuadrada de uno negativo.El discriminante determina la naturaleza de la solución; un discriminante positivo da dos soluciones reales desiguales.Un discriminante cero da una solución real repetida, lo que significa que ambas soluciones de la fórmula cuadrática son iguales.Un discriminante negativo da dos soluciones complejas: conjugados no reales y complejos, es decir, pares de números complejos con partes reales iguales y partes imaginarias con signo opuesto.Un ejemplo de tal ecuación cuadrática tiene soluciones que incluyen la unidad imaginaria i, resultante de un discriminante negativo.Sustituir estas soluciones complejas en la ecuación original da los mismos valores en ambos lados, lo que confirma su validez.Considere un rectángulo con un perímetro de 20 cm, área de 30 cm². Sustituir la ecuación perimetral en la expresión de área da una ecuación cuadrática con un discriminante negativo, lo que lleva a raíces complejas, lo que muestra que no puede existir un rectángulo real.

ecuaciones cuadráticas en el sistema de números complejos

Quadratic Equations in the Complex Number System

Una ecuaci&#243;n cuadr&#225;tica de la forma ax^2+bx+c=0 puede tener soluciones cuya naturaleza var&#237;a seg&#250;n el valor del discriminante, b^2-4ac. En esta expresi&#243;n, a es el coeficiente del t&#233;rmino cuadr&#225;tico x^2, b es el coeficiente del t&#233;rmino lineal x y c es el t&#233;rmino constante. Cuando el discriminante es negativo, la ecuaci&#243;n no tiene soluciones reales. Sin embargo, al introducir los n&#250;meros complejos mediante la unidad imaginaria i, definida por i=&#8730;(-1), estas ecuaciones pueden seguir resolvi&#233;ndose.La ra&#237;z cuadrada de un n&#250;mero negativo se reescribe mediante i. Por ejemplo, en lugar de escribir &#8730;(-36), se expresa como &#8730;36&#183;i=6i. Este ajuste permite representar las soluciones como n&#250;meros complejos, que tienen componente real y componente imaginaria.Para ver c&#243;mo funciona en contexto, considere la siguiente ecuaci&#243;n cuadr&#225;tica x^2+6x+13=0. Los coeficientes aqu&#237; son a=1, b=6 y c=13. Antes de aplicar la f&#243;rmula general, conviene examinar el discriminante:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1383/85;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_1_editor_40033.svg' alt='40033_Equation_1.svg' /></figure>Como el discriminante es negativo, la ra&#237;z cuadrada de -16 se escribe como 4i. Sustituyendo en la f&#243;rmula general,<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:55.45%;'><img style='aspect-ratio:1785/202;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_2_editor_40033.svg' alt='40033_Equation_2.svg' /></figure>Las soluciones, -3+2i y -3-2i, forman un par de n&#250;meros complejos conjugados.Sustituir una de ellas en la ecuaci&#243;n original ayuda a confirmar su validez. Usando -3+2i, el lado izquierdo se convierte en:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:891/85;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_3_editor_40033.svg' alt='40033_Equation_3.svg' /></figure>Al desarrollar y simplificar esta expresi&#243;n se obtiene 0, lo que confirma que el valor satisface la ecuaci&#243;n.Este ejemplo muestra c&#243;mo los n&#250;meros complejos ampl&#237;an el sistema num&#233;rico para garantizar que todas las ecuaciones cuadr&#225;ticas tengan soluci&#243;n. El discriminante sigue guiando la naturaleza de las soluciones: positivo para soluciones reales distintas, cero para una soluci&#243;n real doble y negativo para pares de n&#250;meros complejos conjugados.

Ecuaciones cuadráticas en el sistema de números complejos

Una ecuaci&#243;n cuadr&#225;tica de la forma ax^2+bx+c=0 puede tener soluciones cuya naturaleza var&#237;a seg&#250;n el valor del discriminante, ...

quadratic-equations-in-the-complex-number-system

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complex roots of quadratic equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect.

When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a negative square root can be rewritten using i. For example, √−36 becomes 6i.

This makes it possible to write solutions as complex numbers, which have both a real part and an imaginary part. Consider the quadratic equation x² + 6x + 13 = 0. Here, a = 1, b = 6, and c = 13. The discriminant is negative, so √−16 is written as 4i before using the quadratic formula.

Substituting into the formula gives the solutions −3 + 2i and −3 − 2i. These form a pair of complex conjugates. One solution can be checked by substituting it back into the original equation. Using −3 + 2i and simplifying gives 0, which confirms the solution.

This example shows how complex numbers extend the number system so quadratic equations can still be solved. The discriminant still tells the type of answer: positive for two distinct real solutions, zero for one repeated real solution, and negative for a complex conjugate pair.

Complex Roots of Quadratic Equations

real number system

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complex numbers

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quadratic equations in the complex number system

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

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key terms and concepts

Learn how the discriminant shows whether a quadratic has real or complex roots, and solve an example using imaginary numbers and conjugates.

Complex Roots of Quadratic Equations | JoVE K12

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Complex Roots of Quadratic Equations

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