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

판별식(2차 공식의 제곱근 아래 표현식)이 음수 값을 갖는 이차 방정식에는 실제 해가 없습니다.이를 해결하기 위해 음수 시스템의 제곱근으로 정의되는 허수 단위 i를 사용하여 복소수를 포함하도록 확장됩니다.판별식은 해의 특성을 결정합니다. 양의 판별식은 두 개의 불평등한 실수 해를 제공합니다.0 판별식은 하나의 반복 실수 해를 제공하며, 이는 2차 공식의 두 해가 동일하다는 것을 의미합니다.음의 판별식은 두 가지 복잡한 해, 즉 비실수 및 복소수 활용체, 즉 동일한 실수 부분을 가진 복소수 쌍과 반대 부호를 가진 허수 쌍을 제공합니다.이러한 이차 방정식의 예에는 음의 판별식에서 발생하는 허수 단위 i를 포함하는 해가 있습니다.이러한 복잡한 해를 원래 방정식에 대입하면 양쪽에 동일한 값이 제공되어 타당성이 확인됩니다.둘레가 20cm이고 면적이 30cm²인 직사각형을 고려하십시오. 둘레 방정식을 면적 표현식에 대입하면 음의 판별식이 있는 이차 방정식이 생성되어 실제 직사각형이 존재할 수 없음을 보여주는 복소수 근이 생성됩니다.

복소수 체계의 이차방정식

Quadratic Equations in the Complex Number System

ax^2+bx+c=0 &#54805;&#53468;&#51032; &#51060;&#52264;&#48169;&#51221;&#49885;&#51008; &#54032;&#48324;&#49885; b^2-4ac&#51032; &#44050;&#50640; &#46384;&#46972; &#54644;&#51032; &#49457;&#51656;&#51060; &#45804;&#46972;&#51665;&#45768;&#45796;. &#50668;&#44592;&#49436; a&#45716; &#51060;&#52264;&#54637; x^2&#51032; &#44228;&#49688;, b&#45716; &#51068;&#52264;&#54637; x&#51032; &#44228;&#49688;, c&#45716; &#49345;&#49688;&#54637;&#51077;&#45768;&#45796;. &#54032;&#48324;&#49885;&#51060; &#51020;&#49688;&#51060;&#47732; &#49892;&#49688;&#54644;&#44032; &#51316;&#51116;&#54616;&#51648; &#50506;&#49845;&#45768;&#45796;. &#44536;&#47084;&#45208; i=&#8730;(-1)&#47196; &#51221;&#51032;&#46104;&#45716; &#54728;&#49688; &#45800;&#50948; i&#47484; &#46020;&#51077;&#54616;&#47732; &#51060;&#47084;&#54620; &#48169;&#51221;&#49885;&#46020; &#54400; &#49688; &#51080;&#49845;&#45768;&#45796;.&#51020;&#49688;&#51032; &#51228;&#44273;&#44540;&#51008; i&#47484; &#51060;&#50857;&#54644; &#45796;&#49884; &#50424; &#49688; &#51080;&#49845;&#45768;&#45796;. &#50696;&#47484; &#46308;&#50612; &#8730;(-36)&#51008; &#8730;36&#215;i=6i&#47196; &#54364;&#54788;&#54633;&#45768;&#45796;. &#51060;&#47084;&#54620; &#54364;&#44592;&#47484; &#53685;&#54644; &#54644;&#47484; &#49892;&#49688;&#48512;&#50752; &#54728;&#49688;&#48512;&#47484; &#54632;&#44760; &#44032;&#51652; &#48373;&#49548;&#49688;&#47196; &#45208;&#53440;&#45244; &#49688; &#51080;&#49845;&#45768;&#45796;.&#51060;&#47084;&#54620; &#50896;&#47532;&#44032; &#50612;&#46523;&#44172; &#51201;&#50857;&#46104;&#45716;&#51648; &#48372;&#44592; &#50948;&#54644; &#51060;&#52264;&#48169;&#51221;&#49885; x^2+6x+13=0&#51012; &#49332;&#54196;&#48372;&#44192;&#49845;&#45768;&#45796;. &#50668;&#44592;&#49436; &#44228;&#49688;&#45716; &#44033;&#44033; a=1, b=6, c=13&#51077;&#45768;&#45796;. &#44540;&#51032; &#44277;&#49885;&#51012; &#51201;&#50857;&#54616;&#44592; &#51204;&#50640; &#54032;&#48324;&#49885;&#51012; &#47676;&#51200; &#44160;&#53664;&#54616;&#47732; &#45796;&#51020;&#44284; &#44057;&#49845;&#45768;&#45796;:<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>&#54032;&#48324;&#49885;&#51060; &#51020;&#49688;&#51060;&#48064;&#47196; &#8730;(-16)&#51008; 4i&#47196; &#50424; &#49688; &#51080;&#49845;&#45768;&#45796;. &#44540;&#51032; &#44277;&#49885;&#50640; &#45824;&#51077;&#54616;&#47732;,<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>&#54644; -3+2i&#50752; -3-2i&#45716; &#49436;&#47196; &#53028;&#47112;&#48373;&#49548;&#49688;&#51064; &#54620; &#49933;&#51077;&#45768;&#45796;.&#51060; &#44032;&#50868;&#45936; &#54616;&#45208;&#47484; &#50896;&#47000; &#48169;&#51221;&#49885;&#50640; &#45824;&#51077;&#54616;&#47732; &#48169;&#51221;&#49885;&#51060; &#52280;&#51076;&#51012; &#54869;&#51064;&#54624; &#49688; &#51080;&#49845;&#45768;&#45796;. -3+2i&#47484; &#49324;&#50857;&#54616;&#47732; &#51340;&#48320;&#51008; &#45796;&#51020;&#44284; &#44057;&#49845;&#45768;&#45796;:<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>&#51060; &#49885;&#51012; &#51204;&#44060;&#54616;&#50668; &#51221;&#47532;&#54616;&#47732; 0&#51060; &#46104;&#50612;, &#54644;&#45817; &#44050;&#51060; &#48169;&#51221;&#49885;&#51012; &#47564;&#51313;&#54632;&#51012; &#54869;&#51064;&#54624; &#49688; &#51080;&#49845;&#45768;&#45796;.&#51060; &#50696;&#45716; &#48373;&#49548;&#49688;&#44032; &#49688; &#52404;&#44228;&#47484; &#54869;&#51109;&#54616;&#50668; &#47784;&#46304; &#51060;&#52264;&#48169;&#51221;&#49885;&#50640; &#54644;&#44032; &#51316;&#51116;&#54616;&#46020;&#47197; &#54632;&#51012; &#48372;&#50668; &#51469;&#45768;&#45796;. &#54032;&#48324;&#49885;&#51008; &#54644;&#51032; &#49457;&#51656;&#51012; &#54032;&#45800;&#54616;&#45716; &#51648;&#52840;&#51004;&#47196;&#49436;, &#44050;&#51060; &#50577;&#49688;&#51060;&#47732; &#49436;&#47196; &#45796;&#47480; &#46160; &#49892;&#49688;&#54644;, 0&#51060;&#47732; &#51473;&#44540;, &#51020;&#49688;&#51060;&#47732; &#53028;&#47112;&#48373;&#49548;&#49688; &#54620; &#49933;&#51060; &#46121;&#45768;&#45796;.

ax^2+bx+c=0 &#54805;&#53468;&#51032; &#51060;&#52264;&#48169;&#51221;&#49885;&#51008; &#54032;&#48324;&#49885; b^2-4ac&#51032; &#44050;&#50640; ...

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

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Foundations of Mathematics

수학의 기초

KEY TERMS AND CONCEPTS

주요 용어 및 개념

복소수 체계의 이차 방정식

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...

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radicals

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Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

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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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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