Question

In Problems 59-78, solve each equation in the complex number system. $$ x^{2}+4=0 $$

Short answer

The solutions are \( x = 2i \) and \( x = -2i \).

Step-by-step solution

Rewrite the Equation

Start by rewriting the given equation to isolate the term with the variable. The equation is given as \[ x^2 + 4 = 0 \]. Subtract 4 from both sides to isolate the quadratic term: \[ x^2 = -4 \].

Take the Square Root of Both Sides

Take the square root of both sides of the equation to solve for \(x\): \[ x = \pm \sqrt{-4} \].

Simplify the Square Root of a Negative Number

Recall that the square root of a negative number involves the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \[ x = \pm \sqrt{-4} = \pm \sqrt{4 \cdot (-1)} = \pm 2i \].

Key concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. Traditionally, they are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This form is essential for solving equations that involve the square root of negative numbers.
For instance, in the equation given in the problem, solving for \(x\) results in \(x = \pm 2i\). Here, 0 is the real part, and \(2i\) is the imaginary part, indicating that the solutions are purely imaginary numbers. Complex numbers are fundamental in many areas of mathematics, physics, and engineering.
They allow us to perform operations that would not be possible with real numbers alone, especially in solving polynomial equations that do not have real solutions.

Square Root of Negative Numbers

The square root of a negative number cannot be addressed using only real numbers. The square root function, when applied to a negative number, results in an imaginary number. For instance, the square root of -4 is not defined within the set of real numbers, as there is no real number that satisfies the equation \(x^2 = -4\).
To deal with such situations, we introduce the concept of imaginary numbers. By definition, \(\sqrt{-1} = i\), the imaginary unit. This allows us to express the square root of any negative number using \(i\).
For example, \(\sqrt{-4} = \sqrt{4 \cdot (-1)} = 2i\). By using the properties of the imaginary unit, we can simplify the square roots of negative numbers into a form that is more manageable and useful for further calculations.

Imaginary Unit i

The imaginary unit, denoted by \(i\), is a mathematical tool used to handle the square root of negative numbers. By definition, \(i = \sqrt{-1}\). The introduction of \(i\) allows us to extend the real number system to include complex numbers. Here are some key properties of \(i\):

• \(i^2 = -1\)
• \(i^3 = i \cdot i^2 = i \cdot -1 = -i\)
• \(i^4 = (i^2)^2 = (-1)^2 = 1\)

These properties repeat cyclically.
Understanding these properties is crucial when working with complex numbers, as they simplify computations significantly. For example, in the problem provided, knowing that \(i^2 = -1\) allows us to express \(\sqrt{-4}\) as \(2i\). This simplifies finding the solutions to the quadratic equation. The imaginary unit \(i\) thus plays a vital role in extending the boundaries of conventional algebra.