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Completing the square is a mathematical technique used primarily to work with quadratic equations. This method transforms a quadratic expression or equation into a perfect square trinomial, making it easy to solve or simplify.
In the given exercise, we start with a binomial, which is a polynomial with two terms: \(z^2 - 3z\). The process involves adding a specific constant to complete the square, essentially reshaping the expression into a form where it can be easily squared.

• Identify the coefficient of 'the linear term'. In our example, it's the -3 from \(z^2 - 3z\).
• Divide that coefficient by 2. Here, \(-3/2\) is the result.
• Square this result. The calculation gives \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).
• Add this squared number to the original binomial to complete the square.

This added constant, \(\frac{9}{4}\), completes the square, creating a trinomial that is a perfect square.

A binomial is a type of algebraic expression that consists of exactly two terms. In our problem, the expression \(z^2 - 3z\) is a binomial, consisting of a quadratic term \(z^2\), and a linear term, which is \(-3z\).
Binomials are fundamental building blocks in algebra because many algebraic techniques, like factoring or expansion, involve working with these two-term expressions. Understanding the structure of a binomial is essential when using methods like completing the square.

• Each term in a binomial is separated by a plus \(\,+\) or minus \(\,-\) sign.
• The terms can be constants, variables, or both. For instance, \(z^2\) (a variable term) and \(-3z\) (a variable with a coefficient).
• Binomials can be restructured by adding constants, transforming them into trinomials or higher-degree polynomials if needed.

In this exercise, adding the constant \(\frac{9}{4}\) converts the binomial \(z^2 - 3z\) into a perfect square trinomial.

A squared expression is a mathematical expression raised to the power of two. When working with quadratic expressions, one goal is often to rewrite them as a square of a binomial because squared expressions are simplified and more intuitive to manage.
In this exercise, once the binomial \(z^2 - 3z\) was transformed into a perfect square trinomial \(z^2 - 3z + \frac{9}{4}\), it can then be expressed as a squared expression.
The trinomial \(z^2 - 3z + \frac{9}{4}\) simplifies to \(\left(z - \frac{3}{2}\right)^2\). This form indicates that the expression is a square, showing clearly that it results from multiplying \(z - \frac{3}{2}\) by itself.

• Squared expressions often represent the area of a square with side length equal to the binomial factor.
• They provide a concise and clear method of representing the same numerical value.
• The process allows for easier calculations in solving equations and graphing.

By viewing and rewriting the expression as a squared expression, students can tackle more complex algebra problems with greater ease.