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

Les équations du second degré dans lesquelles le discriminant — l’expression sous la racine carrée dans la formule du second degré — a une valeur négative n’ont pas de solutions réelles.Pour les résoudre, le système de numération est étendu pour inclure les nombres complexes en utilisant l’unité imaginaire i, définie comme la racine carrée d’un moins un.Le discriminant détermine la nature de la solution ; Un discriminant positif donne deux solutions réelles inégales.Un discriminant zéro donne une solution réelle répétée, ce qui signifie que les deux solutions de la formule quadratique sont égales.Un discriminant négatif donne deux solutions complexes : les conjugués non réels et complexes, c’est-à-dire les paires de nombres complexes avec des parties réelles égales et les parties imaginaires avec des signes opposés.Un exemple d’une telle équation quadratique a des solutions incluant l’unité imaginaire i, résultant d’un discriminant négatif.En substituant ces solutions complexes dans l’équation d’origine, on obtient les mêmes valeurs des deux côtés, confirmant leur validité.Considérons un rectangle d’un périmètre de 20 cm, d’une aire de 30 cm². En substituant l’équation du périmètre dans l’expression de l’aire, on obtient une équation du second degré avec un discriminant négatif, ce qui conduit à des racines complexes, montrant qu’aucun rectangle réel ne peut exister.

Équations du second degré dans l’ensemble des nombres complexes

Quadratic Equations in the Complex Number System

Une &#233;quation du second degr&#233; de la forme ax^2 + bx + c = 0 peut admettre des solutions de nature diff&#233;rente selon la valeur du discriminant, b^2 - 4ac. Dans cette expression, a d&#233;signe le coefficient du terme quadratique x^2, b le coefficient du terme lin&#233;aire x et c le terme constant. Lorsque le discriminant est n&#233;gatif, l&#8217;&#233;quation ne poss&#232;de pas de solutions r&#233;elles. Toutefois, en introduisant les nombres complexes &#224; l&#8217;aide de l&#8217;unit&#233; imaginaire i, d&#233;finie par i = &#8730;(-1), ces &#233;quations peuvent &#234;tre r&#233;solues.La racine carr&#233;e d&#8217;un nombre n&#233;gatif s&#8217;exprime alors &#224; l&#8217;aide de i. Par exemple, au lieu d&#8217;&#233;crire &#8730;(-36), on &#233;crit &#8730;36 &#215; i = 6i. Cette transformation permet d&#8217;exprimer les solutions sous forme de nombres complexes, comportant &#224; la fois une partie r&#233;elle et une partie imaginaire.Pour illustrer ce proc&#233;d&#233;, consid&#233;rons l&#8217;&#233;quation du second degr&#233; x^2 + 6x + 13 = 0. Les coefficients sont ici a = 1, b = 6 et c = 13. Avant d&#8217;appliquer la formule du second degr&#233;, il convient d&#8217;examiner le discriminant&#160;:<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>Le discriminant &#233;tant n&#233;gatif, la racine carr&#233;e de -16 s&#8217;&#233;crit 4i. En substituant dans la formule du second degr&#233;&#160;:<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>Les solutions, -3 + 2i et -3 - 2i, forment une paire de conjugu&#233;s complexes.Pour v&#233;rifier la validit&#233; de l&#8217;une de ces solutions, substituons -3 + 2i dans l&#8217;&#233;quation initiale. Le membre de gauche devient&#160;:<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>En d&#233;veloppant et en simplifiant cette expression, on obtient 0, ce qui confirme que cette valeur satisfait bien l&#8217;&#233;quation.Cet exemple met en &#233;vidence la mani&#232;re dont les nombres complexes &#233;tendent l&#8217;ensemble des nombres afin que toute &#233;quation du second degr&#233; admette une solution. Le discriminant continue de d&#233;terminer la nature des solutions : positif pour deux solutions r&#233;elles distinctes, nul pour une solution r&#233;elle double, et n&#233;gatif pour une paire de conjugu&#233;s complexes.

Une &#233;quation du second degr&#233; de la forme ax^2 + bx + c = 0 peut admettre des solutions de nature diff&#233;rente selon la valeur du ...

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

Number Types on the Real Line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

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exponents

Exponent Rules and Scientific Notation

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radicals

Simplifying Radicals

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algebraic expressions

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rational expressions

Working with Rational Expression Domains

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quadratic equations

Solving Quadratics by Factoring and Formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex Numbers in Solving Equations

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...

radical equations

Solving Radical Equations with Checks

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inequalities

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absolute value inequalities

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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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