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When facing a quadratic equation, which is any equation that can be rearranged into the standard form of \( ax^2 + bx + c = 0 \), one of the key strategies to find its solutions is the use of the square root property. But what does this mean exactly?
The square root property comes into play when we have a term squared, as seen in the equation \( (x-3)^2 = -5 \). Its general use in mathematics is to undo the squaring, essentially finding what was originally squared to get our equation.
In practical terms, after isolating the squared term, applying this property means taking the square root of both sides. The goal here is to end up with just \(x \) on one side, but an important detail to remember is that square roots have two possible values: one positive and one negative. This is why we see \( x - 3 = \pm\sqrt{-5} \), to account for both potential solutions.

Imaginary numbers might sound like something out of a fantasy novel, but they are an essential component of higher mathematics, especially when solving quadratic equations with negative values under the square root.
The concept of imaginary numbers arises when trying to calculate the square root of a negative number, something you cannot do with just the real number system. To handle this, mathematicians have introduced the imaginary unit, represented by \(i\), where \( i^2 = -1 \).
So, for example, \( \sqrt{-5} \) becomes \( \sqrt{5} \cdot i \). This is precisely what occurs in the given exercise, transitioning from a seemingly unsolvable square root of a negative number to something manageable using imaginary numbers.

With the introduction of imaginary numbers, we can now solve equations resulting in complex solutions. A complex solution to a quadratic equation includes both a real part and an imaginary part, generally expressed in the form \( a + bi \), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
Here's an interesting part: every quadratic equation has exactly two solutions, which may be real, imaginary, or complex. In our exercise, the solutions are complex because they consist of a real part, which is 3, and an imaginary part, which is \( \pm\sqrt{5} \cdot i \). Therefore, the complete solutions to the given equation are represented as \( x = 3 \pm i\sqrt{5} \), incorporating both of these components.