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

Equações quadráticas nas quais o discriminante - a expressão sob a raiz quadrada na fórmula quadrática - tem um valor negativo não têm soluções reais.Para resolvê-los, o sistema numérico é estendido para incluir números complexos usando a unidade imaginária i, definida como a raiz quadrada de uma negativa.O discriminante determina a natureza da solução; Um discriminante positivo dá duas soluções reais desiguais.Um discriminante zero dá uma solução real repetida, o que significa que ambas as soluções da fórmula quadrática são iguais.Um discriminante negativo dá duas soluções complexas: conjugados não reais e complexos - significando, pares de números complexos com partes reais iguais e partes imaginárias com sinal oposto.Um exemplo de tal equação quadrática tem soluções, incluindo a unidade imaginária i, resultante de um discriminante negativo.Substituir essas soluções complexas na equação original fornece os mesmos valores em ambos os lados, confirmando sua validade.Considere um retângulo com perímetro de 20 cm, área de 30 cm². Substituindo a equação de perímetro na expressão de área, obtém-se uma equação quadrática com um discriminante negativo, levando a raízes complexas, mostrando que nenhum retângulo real pode existir.

equações quadráticas no sistema de números complexos

Quadratic Equations in the Complex Number System

Uma equa&#231;&#227;o quadr&#225;tica na forma ax^2 + bx + c = 0 pode apresentar solu&#231;&#245;es de naturezas distintas, dependendo do valor do discriminante, b^2 - 4ac. Nessa express&#227;o, a &#233; o coeficiente do termo quadr&#225;tico x^2, b &#233; o coeficiente do termo linear x e c &#233; o termo constante. Quando o discriminante &#233; negativo, a equa&#231;&#227;o n&#227;o possui solu&#231;&#245;es no conjunto dos n&#250;meros reais. Contudo, ao introduzir os n&#250;meros complexos por meio da unidade imagin&#225;ria i, definida por i = &#8730;(-1), essas equa&#231;&#245;es podem ser resolvidas.A raiz quadrada de um n&#250;mero negativo &#233; reescrita com o uso de i. Por exemplo, em vez de escrever &#8730;(-36), ela &#233; expressa como &#8730;36 &#215; i = 6i. Esse procedimento permite representar as solu&#231;&#245;es como n&#250;meros complexos, que possuem partes real e imagin&#225;ria.Para observar como isso se aplica na pr&#225;tica, considere a equa&#231;&#227;o quadr&#225;tica x^2 + 6x + 13 = 0. Os coeficientes s&#227;o a = 1, b = 6 e c = 13. Antes de aplicar a f&#243;rmula quadr&#225;tica, &#233; conveniente analisar o 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 o discriminante &#233; negativo, a raiz quadrada de -16 &#233; escrita como 4i. Substituindo na f&#243;rmula quadr&#225;tica:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:55.96%;'><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>As solu&#231;&#245;es -3 + 2i e -3 - 2i formam um par de n&#250;meros complexos conjugados.A substitui&#231;&#227;o de uma dessas solu&#231;&#245;es na equa&#231;&#227;o original ajuda a confirmar sua validade. Utilizando -3 + 2i, o lado esquerdo torna-se:<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>Ao expandir e simplificar essa express&#227;o, obt&#233;m-se 0, o que confirma que o valor satisfaz a equa&#231;&#227;o.Este exemplo evidencia como os n&#250;meros complexos ampliam o sistema num&#233;rico, garantindo que todas as equa&#231;&#245;es quadr&#225;ticas possuam solu&#231;&#245;es. O discriminante continua a determinar a natureza das solu&#231;&#245;es: positivo para solu&#231;&#245;es reais distintas, zero para uma solu&#231;&#227;o real dupla e negativo para pares de n&#250;meros complexos conjugados.

Equações Quadráticas no Sistema de Números Complexos

Uma equa&#231;&#227;o quadr&#225;tica na forma ax^2 + bx + c = 0 pode apresentar solu&#231;&#245;es de naturezas distintas, dependendo do valor do ...

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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

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complex numbers

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Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

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

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Complex Roots of Quadratic Equations

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