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

Quadratic equations in which the discriminant &#8212;the expression under the square root in the quadratic formula&#8212;has a negative value have no real solutions.To solve these, the number system is extended to include complex numbers using the imaginary unit i, defined as the square root of a negative one.The discriminant determines the solution's nature; a positive discriminant gives two unequal real solutions.A zero discriminant gives one repeated real solution, meaning both solutions from the quadratic formula are equal.A negative discriminant gives two complex solutions: non-real and complex conjugates&#8212;meaning, complex number pairs with equal real parts and imaginary parts with opposite sign.An example of such a quadratic equation has solutions including the imaginary unit i, resulting from a negative discriminant.Substituting these complex solutions into the original equation gives same values on both sides, confirming their validity.Consider a rectangle with a perimeter of 20 cm, area of 30 cm&#178;. Substituting the perimeter equation into the area expression gives a quadratic equation with a negative discriminant, leading to complex roots&#8212;showing no real rectangle can exist.

quadratic equations in the complex number system

A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2&#8722;4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of a negative number is rewritten using i. For instance, instead of writing &#8722;36, it is expressed as 36i=6i. This adjustment allows solutions to be represented as complex numbers, which have both real and imaginary components.To see how this works in context, consider the quadratic equation x2+6x+13=0 . The coefficients here are a=1, b=6, and c=13. Before applying the quadratic formula, it helps to examine the discriminant:<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>Because the discriminant is negative, the square root of -16 is written as 4i. Substituting into the quadratic formula,<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:55.41%;'><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>The solutions, &#8722;3+2i and &#8722;3-2i, form a pair of complex conjugates.Substituting one of these back into the original equation helps confirm its validity. Using &#8722;3+2i, the left-hand side becomes:<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>Expanding and simplifying this expression results in 0, verifying that the value satisfies the equation.This example highlights how complex numbers extend the number system to ensure all quadratic equations have solutions. The discriminant continues to guide the nature of the solutions: positive for distinct real solutions, zero for a repeated real solution, and negative for complex conjugate pairs.

Quadratic Equations in the Complex Number System

A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2&#8722;4ac. In this ...

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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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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

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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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