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

判别式（二次公式中平方根下的表达式）为负值的二次方程没有实数解。为了解决这些问题，数系被扩展为包含复数，该数系引入了虚数单位 i，其定义为负一的平方根。判别式决定了方程解的性质；当判别式为正时，方程有两个不相等的实数解。判别式为零时，得到一个重根实数解，即二次公式给出的两个解相等。负的判别式给出两个复数解：非实数且互为共轭复数——即实部相等、虚部符号相反的复数对。此类二次方程的一个例子，其解包含虚数单位 i，这是由负判别式导致的结果。将这些复杂解代入原方程后，等式两边得到相同的值，从而验证了其正确性。考虑一个周长为 20 cm、面积为 30 cm² 的矩形。将周长方程代入面积表达式后，得到一个判别式为负数的二次方程，其根为复数——这表明不存在满足条件的实矩形。

复数数域中的二次方程

Quadratic Equations in the Complex Number System

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&#24418;&#22914; ax^2+bx+c=0 ...

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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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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