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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand various relationships. In algebra, you will often encounter expressions, equations, and functions that represent real-world problems in a more abstract way.

For instance, when we have a binomial expression, such as \((a - d)\), it consists of two terms subtracted from each other, 'a' and 'd'. Algebra allows us to perform operations on this expression, like squaring it, to see how it behaves and to simplify it for further calculations. It's critical in algebra to understand how to handle and transform these expressions using established rules and formulas, which will be the focus of the following sections.

The binomial expansion is a way to express the power of a binomial, which is an algebraic expression containing two terms. For example, \((a - d)\) is a binomial. When we raise it to a power, such as squaring it, we are expanding this binomial.

The expansion involves using the powers of each term and the coefficient in a particular pattern, which we can predict using Pascal's Triangle or the Binomial Theorem for larger exponents. In the case of squaring a binomial, the pattern follows the squaring rule. It's important to understand that during the expansion, the middle term comes from the product of the two terms in the binomial multiplied by 2. This illustrates why, when expanding \((a - d)^2\), we get \(a^2 - 2ad + d^2\), not just squaring each term on its own.

The squaring rule is a specific application of binomial expansion. It enables us to square a binomial quickly and accurately without having to distribute the multiplication over each term. According to the squaring rule, \((a \pm b)^2 = a^2 \pm 2ab + b^2\). Here, the '\(\pm\)' symbol indicates that the rule applies whether the terms are being added or subtracted.

Let's break it down: the first term of the result, \(a^2\), comes from squaring the first term of the binomial. The last term, \(b^2\), comes from squaring the second term. The middle term, which is twice the product of the first and second terms (\(2ab\)), is what many students forget. Remembering to include it is vital for correctly expanding the binomial. This rule is a cornerstone of algebra because it simplifies and speeds up the process of dealing with squared expressions.