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In mathematics, imaginary numbers extend the concept of real numbers by including the square root of negative numbers. This is vital as it permits solutions to certain equations where real numbers cannot suffice. An imaginary number is represented by the symbol \( i \), which is defined as \( i = \sqrt{-1} \). Hence, any imaginary number can be written in the form \( bi \), where \( b \) is a real number.

Imaginary numbers come into play in various mathematical areas, particularly in complex numbers and other advanced fields. For example, consider \( \sqrt{-6} \). To express this using imaginary numbers, rewrite it as \( \sqrt{6} * i \). This expression demonstrates how we can handle negative square roots by leveraging imaginary numbers. They're an essential tool for transforming seemingly unsolvable problems into manageable ones using complex numbers.

Binomial expansion is a mathematical technique used to expand expressions that are raised to powers. One of the most common forms is the binomial square formula, \((a - b)^2 = a^2 - 2ab + b^2\). This formula helps to simplify expressions that involve sums or differences squared.

Let's illustrate this with the complex number expression \((2 - \sqrt{6}i)^2\). Here, it acts as our 'binomial' with \(a = 2\) and \(b = \sqrt{6}i\). Expanding it by the binomial formula gives, \(2^2 - 2*2*\sqrt{6}i + (\sqrt{6}i)^2\), resulting in \(4 - 4\sqrt{6}i - 6\).

Understanding binomial expansion aids drastically in simplifying and calculating powers of complex numbers, which then makes it possible to further process them into standard form.

The standard form of complex numbers is written as \( a + bi \), where \(a\) and \(b\) are real numbers. This form neatly separates the real part \(a\) from the imaginary part \(bi\), providing clarity and ease in mathematical operations.

Returning to our earlier focus on \((2 - \sqrt{6}i)^2\), after employing binomial expansion, the expression simplifies to \(-2 - 4\sqrt{6}i\). This fits the standard form \( a + bi \), where the real part \( a = -2 \) and the imaginary part is \( bi = -4\sqrt{6}i \).

Understanding how to express complex numbers in standard form is crucial for their manipulation and use in equations both practical and theoretical. This structured format makes complex operations like addition, subtraction, and multiplication straightforward and consistent.