91Ó°ÊÓ

Imaginary numbers are a core concept in higher mathematics, especially when dealing with quadratic equations with complex solutions. An imaginary number is defined as a number that can be written as a real number multiplied by the imaginary unit \(i\). The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This means that the imaginary unit is the square root of -1. This concept is crucial because it allows us to find solutions to equations that do not have real solutions. For example, in the given exercise, we encounter the term \(\sqrt{-4}\), which is not a real number. By recognizing that \(\sqrt{-4}\) can be written as \(\sqrt{4} \, \sqrt{-1}\), or simply \(2i\), we can solve the equation within the realm of complex numbers.

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They can have real or complex solutions based on the discriminant \(b^2 - 4ac\). When the discriminant is negative, the quadratic equation has complex solutions, which involve imaginary numbers.
To solve the given quadratic equation \(x^2 + 4 = 0\), we follow these steps:

• Step 1: Recognize the quadratic equation: \(x^2 + 4 = 0\).
• Step 2: Isolate the \(x^2\) term by subtracting 4 from both sides: \(x^2 = -4\).
• Step 3: Take the square root of both sides. Remember, taking the square root of -4 involves the imaginary unit \(i\): \(x = \pm \sqrt{-4} = \pm 2i\).
• Step 4: State the solutions: \(x = 2i\) and \(x = -2i\).

By following these steps, we find that the solutions to \(x^2 + 4 = 0\) are complex solutions involving imaginary numbers.

Complex solutions arise when solving quadratic equations with a negative discriminant. A complex solution has both a real part and an imaginary part, and it is generally expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. However, in some cases, like our exercise, the real part \(a\) might be zero.
For the equation \(x^2 + 4 = 0\), the complex solutions are \(x = 2i\) and \(x = -2i\). Here:

• The real part \(a\) is 0.
• The imaginary part \(bi\) is \(2i\) or \(-2i\).

These solutions clearly illustrate how complex numbers can solve equations that would otherwise have no real solutions. Understanding these concepts is important, as complex solutions frequently appear in various fields of science, engineering, and mathematics.