Introduction

Quadratic equations are a fundamental part of mathematics and have wide-ranging applications in various fields such as physics, engineering, and economics. Traditionally, these equations are solved using methods such as factoring or the quadratic formula. However, the solution set of a quadratic equation expands to include complex numbers when the discriminant is negative. This article will explore how to solve quadratic equations in the complex number system, a topic that is particularly relevant for students and professionals in mathematics and related fields.

Understanding Quadratic Equations in the Complex Number System

A quadratic equation is generally written in the form ax2 bx c 0. When the discriminant (Δ b2 - 4ac) is negative, the equation does not have real solutions but instead has complex conjugate solutions. This is because, in the complex number system, every element has a square root, allowing us to extend the field of real numbers to the complex numbers.

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. For a quadratic equation of the form ax2 bx c 0, the solutions can be found using the formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Example: Solving a Quadratic Equation with Complex Solutions

Let's solve the equation x2 5x 7 0.

Step 1: Identifying Coefficients

The coefficients are: a 1, b 5, and c 7.

Step 2: Calculating the Discriminant

The discriminant Δ is calculated as follows:

Δ b2 - 4ac 52 - 4 cdot 1 cdot 7 25 - 28 -3

Since the discriminant is negative, the equation has complex solutions.

Step 3: Finding the Solutions

The solutions are given by:

(x_{1} frac{-5 sqrt{-3}}{2} frac{-5 isqrt{3}}{2})

(x_{2} frac{-5 - sqrt{-3}}{2} frac{-5 - isqrt{3}}{2})

So, the roots are (frac{-5 isqrt{3}}{2}) and (frac{-5 - isqrt{3}}{2}).

Complex Numbers and the Quadratic Formula

It's important to note that the quadratic formula can be applied to equations with complex coefficients without extension of the field. For instance, consider the equation x2 - 2ix - 2i 0. Here, the coefficients are complex numbers, but the method remains the same.

Using the quadratic formula:

(x frac{2i pm sqrt{(-2i)^2 - 4 cdot 1 cdot (-2i)}}{2 cdot 1})

(x frac{2i pm sqrt{-4 - (-8i)}}{2})

(x frac{2i pm sqrt{8i - 4}}{2})

(x frac{2i pm sqrt{4(2i - 1)}}{2})

(x frac{2i pm 2isqrt{2i - 1}}{2})

(x i pm isqrt{2i - 1})

Now, solving the square root of the complex number 2i - 1. Let's use the method of finding square roots of complex numbers:

Let (s sqrt{2i - 1})

(s pm sqrt{sqrt{(2)^2 (-1)^2}} pm sqrt{sqrt{4 1}} pm sqrt{sqrt{5}})

The argument of the complex number is:

(theta tan^{-1}left(frac{-1}{2}right))

Using the polar form, we find:

(s pm left(sqrt{frac{1}{sqrt{5}}} cos frac{theta}{2} - i sqrt{frac{1}{sqrt{5}}} sin frac{theta}{2}right))

Since (s 2 - i), we have:

(x i pm (2 - i) 2, i)

The solutions are (x 2) and (x i).

Conclusion

In conclusion, solving quadratic equations in the complex number system requires understanding the quadratic formula and the properties of complex numbers. By following the steps outlined in this article, you can effectively solve such equations, expanding your mathematical toolkit and deepening your understanding of algebraic concepts.

Related Keywords

quadratic equations, complex numbers, complex solutions