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

Le equazioni di secondo grado in cui il discriminante — l'espressione sotto la radice quadrata nella formula di secondo grado — ha un valore negativo non hanno soluzioni reali.Per risolverli, il sistema numerico viene esteso per includere numeri complessi utilizzando l'unità immaginaria i, definita come la radice quadrata di una negativa.La discriminante determina la natura della soluzione; Un discriminante positivo dà due soluzioni reali disuguali.Un discriminante zero dà una soluzione reale ripetuta, il che significa che entrambe le soluzioni della formula quadratica sono uguali.Un discriminante negativo fornisce due soluzioni complesse: coniugati non reali e complessi, ovvero coppie di numeri complessi con parti reali uguali e parti immaginarie con segno opposto.Un esempio di tale equazione quadratica ha soluzioni che includono l'unità immaginaria i, risultante da un discriminante negativo.Sostituendo queste soluzioni complesse nell'equazione originale si ottengono gli stessi valori su entrambi i lati, confermando la loro validità.Si consideri un rettangolo con un perimetro di 20 cm, un'area di 30 cm². Sostituendo l'equazione perimetrale nell'espressione dell'area si ottiene un'equazione quadratica con un discriminante negativo, che porta a radici complesse, dimostrando che non può esistere un vero rettangolo.

equazioni di secondo grado nel sistema dei numeri complessi

Quadratic Equations in the Complex Number System

Un'equazione di secondo grado della forma ax^2 + bx + c = 0 pu&#242; avere soluzioni di natura diversa a seconda del valore del discriminante, b^2 - 4ac. In questa espressione, a &#232; il coefficiente del termine di secondo grado x^2, b &#232; il coefficiente del termine di primo grado x e c &#232; il termine noto. Quando il discriminante &#232; negativo, l'equazione non ha soluzioni reali. Tuttavia, introducendo i numeri complessi attraverso l'unit&#224; immaginaria i, definita da i = &#8730;(-1), queste equazioni possono comunque essere risolte.La radice quadrata di un numero negativo viene riscritta usando i. Ad esempio, invece di scrivere &#8730;(-36), si scrive &#8730;36&#215;i=6i. Questa modifica consente di rappresentare le soluzioni come numeri complessi, dotati di una parte reale e di una parte immaginaria.Per comprendere come funziona nel contesto, consideriamo l'equazione di secondo grado x^2 + 6x + 13 = 0. I coefficienti qui sono a = 1, b = 6 e c = 13. Prima di applicare la formula risolutiva, &#232; utile esaminare il 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>Poich&#233; il discriminante &#232; negativo, la radice quadrata di -16 si scrive come 4i. Sostituendo nella formula risolutiva:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><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>Le soluzioni, -3 + 2i e -3 - 2i, formano una coppia di coniugati complessi.Sostituendo una di queste soluzioni nell'equazione originale, se ne conferma la validit&#224;. Utilizzando -3 + 2i, il membro sinistro diventa:<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>Sviluppando e semplificando questa espressione, si ottiene 0, verificando che tale valore soddisfa l'equazione.Questo esempio evidenzia come i numeri complessi estendano il sistema numerico per garantire che tutte le equazioni di secondo grado abbiano soluzioni. Il discriminante continua a determinare la natura delle soluzioni: positivo per soluzioni reali distinte, zero per una soluzione reale doppia e negativo per coppie di coniugati complessi.

Equazioni di secondo grado nel sistema dei numeri complessi

Un'equazione di secondo grado della forma ax^2 + bx + c = 0 pu&#242; avere soluzioni di natura diversa a seconda del valore del discriminante, b^2 - 4ac. ...

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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 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

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Complex Roots of Quadratic Equations

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