91Ó°ÊÓ

When working with quadratic equations, a core concept is isolating the square term. This is essentially reorganizing the equation to have the term with the variable squared (e.g., \(x^2\) or \( (x+2)^2 \) ) by itself on one side of the equation. Isolation is critical for simplifying the equation to a form where we can apply further operations, such as extracting square roots.

For the given exercise, isolating the square term involved moving the constant term to the opposite side by subtracting 5 from both sides of the equation. The equation \( (x + 2)^2 + 5 = 0 \) became \( (x + 2)^2 = -5 \) upon isolation of the square term.

This step is straightforward but crucial. It sets the stage for further simplifications and solving the rest of the quadratic equation accurately.

The concept of extracting square roots arises after isolating the square term in a quadratic equation. To find the values of \(x\), we need to perform the operation that is the inverse of squaring, which is the square root.

However, a significant point of caution is that when you take the square root of both sides of an equation, you must include both the positive and negative roots, denoted by \(\pm\). This is because if \(x^2 = a\), then \(x\) could be \(\sqrt{a}\) or \( -\sqrt{a}\) since squaring either the positive or negative root gives the same result.

When performing this step in the solution, it led to \( x + 2 = \pm \sqrt{-5} \), introducing a new component: the concept of square roots of negative numbers, which as we know, are complex numbers.

The equation from the exercise ventured into the realm of complex numbers when extracting the square root of a negative number. Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined as \(i^2 = -1\).

The significance of this is profound as it extends the real number system to include solutions to equations like \(x^2 = -1\), which have no real solutions. They are essential in mathematics and many domains of science and engineering because they provide a complete structure for solving polynomial equations.

In the context of our problem, after extracting the square root, recognizing that \(\sqrt{-5}\) is a complex number led to writing it as \(\sqrt{5}i\). Thus, we find that the solution to the equation \(x = -2 \pm \sqrt{5}i\) includes both the real component, -2, and the imaginary component, \(\pm\sqrt{5}i\).