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

Równania kwadratowe, w których wyróżnik — wyrażenie pod pierwiastkiem kwadratowym we wzorze kwadratowym — mają wartość ujemną, nie mają rzeczywistych rozwiązań.Aby je rozwiązać, system liczbowy jest rozszerzany o liczby zespolone za pomocą wyimaginowanej jednostki i, zdefiniowanej jako pierwiastek kwadratowy z ujemnej.Osoba wyróżniająca określa charakter rozwiązania; Pozytywny dyskryminator daje dwa nierówne realne rozwiązania.Wyróżnik zerowy daje jedno powtarzające się rozwiązanie rzeczywiste, co oznacza, że oba rozwiązania ze wzoru kwadratowego są równe.Negatywny wyróżnik daje dwa złożone rozwiązania: sprzężone nierzeczywiste i zespolone - czyli pary liczb zespolonych z równymi częściami rzeczywistymi i częściami urojonymi o przeciwnym znaku.Przykładem takiego równania kwadratowego są rozwiązania obejmujące urojoną jednostkę i, wynikającą z ujemnego wyróżnika.Podstawienie tych złożonych rozwiązań do oryginalnego równania daje te same wartości po obu stronach, potwierdzając ich słuszność.Rozważ prostokąt o obwodzie 20 cm, powierzchni 30 cm². Podstawienie równania obwodu do wyrażenia powierzchni daje równanie kwadratowe z ujemnym wyróżnikiem, co prowadzi do pierwiastków zespolonych, co pokazuje, że żaden rzeczywisty prostokąt nie może istnieć.

równania kwadratowe w układzie liczb zespolonych

Quadratic Equations in the Complex Number System

R&#243;wnanie kwadratowe w postaci ax^2+bx+c=0 mo&#380;e mie&#263; rozwi&#261;zania o r&#243;&#380;nym charakterze w zale&#380;no&#347;ci od warto&#347;ci wyr&#243;&#380;nika b^2-4ac. W tym wyra&#380;eniu a jest wsp&#243;&#322;czynnikiem przy wyrazie kwadratowym x^2, b jest wsp&#243;&#322;czynnikiem przy wyrazie liniowym x, a c jest wyrazem sta&#322;ym. Gdy wyr&#243;&#380;nik jest ujemny, r&#243;wnanie nie ma rozwi&#261;za&#324; w zbiorze liczb rzeczywistych. Jednak wprowadzaj&#261;c liczby zespolone poprzez jednostk&#281; urojon&#261;, zdefiniowan&#261; jako i=&#8730;(-1), r&#243;wnania te nadal mo&#380;na rozwi&#261;za&#263;.Pierwiastek kwadratowy z liczby ujemnej zapisuje si&#281; z u&#380;yciem i. Na przyk&#322;ad zamiast &#8730;(-36) zapisujemy &#8730;36&#215;i=6i. Ta modyfikacja pozwala przedstawia&#263; rozwi&#261;zania jako liczby zespolone, maj&#261;ce zar&#243;wno sk&#322;adow&#261; rzeczywist&#261;, jak i urojon&#261;.Aby zobaczy&#263;, jak to dzia&#322;a w praktyce, rozwa&#380;my r&#243;wnanie kwadratowe x^2+6x+13=0. Wsp&#243;&#322;czynniki tutaj to a=1, b=6 i c=13. Przed zastosowaniem wzoru kwadratowego warto zbada&#263; wyr&#243;&#380;nik:<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>Poniewa&#380; wyr&#243;&#380;nik jest ujemny, pierwiastek z -16 zapisujemy jako 4i. Podstawiaj&#261;c to do wzoru kwadratowego,<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:55.46%;'><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>Otrzymujemy rozwi&#261;zania -3+2i i -3-2i, kt&#243;re tworz&#261; par&#281; sprz&#281;&#380;onych liczb zespolonych.Podstawienie jednego z nich z powrotem do pierwotnego r&#243;wnania pomaga potwierdzi&#263; poprawno&#347;&#263; rozwi&#261;zania. U&#380;ywaj&#261;c -3+2i, lewa strona przyjmuje posta&#263;:<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>Rozwini&#281;cie i uproszczenie tego wyra&#380;enia daje 0, co potwierdza, &#380;e warto&#347;&#263; spe&#322;nia r&#243;wnanie.Ten przyk&#322;ad pokazuje, jak liczby zespolone rozszerzaj&#261; uk&#322;ad liczbowy, zapewniaj&#261;c istnienie rozwi&#261;za&#324; wszystkich r&#243;wna&#324; kwadratowych. Wyr&#243;&#380;nik nadal okre&#347;la charakter rozwi&#261;za&#324;: dodatni &#8212; dwa r&#243;&#380;ne rozwi&#261;zania rzeczywiste, r&#243;wny zeru &#8212; podw&#243;jny pierwiastek rzeczywisty, ujemny &#8212; para sprz&#281;&#380;onych liczb zespolonych.

Równania kwadratowe w układzie liczb zespolonych

R&#243;wnanie kwadratowe w postaci ax^2+bx+c=0 mo&#380;e mie&#263; rozwi&#261;zania o r&#243;&#380;nym charakterze w zale&#380;no&#347;ci od warto&#347;ci ...

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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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.

math fundamentals

Complex Roots of Quadratic Equations

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