91Ó°ÊÓ

Let's begin with understanding a perfect square trinomial. A perfect square trinomial takes the form of \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). These trinomials can be factored into \((a - b)^2\) or \((a + b)^2\) respectively. Notice how the given quadratic expression in the problem, \(16x^2 - 24xy + 9y^2 - 64\), includes the terms \(16x^2\), \(-24xy\), and \(9y^2\). These correspond to \(a^2\), \(-2ab\), and \(b^2\) respectively, where \(a = 4x\) and \(b = 3y\).

We recognize that:

• \(16x^2\) is \((4x)^2\)
• \(9y^2\) is \((3y)^2\)
• \(-24xy\) is precisely \(-2 \cdot 4x \cdot 3y\)

Thus, the trinomial \(16x^2 - 24xy + 9y^2\) is indeed a perfect square trinomial, and can be factored as \((4x - 3y)^2\).

Now, let's discuss the difference of squares. This is a common algebraic pattern where we have expressions like \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). In the given problem, after factoring the perfect square trinomial, we substitute and obtain \((4x - 3y)^2 - 64\).

Recognizing the term \(64\) as \(8^2\), we see that:

• \( (4x - 3y)^2 \) is \(a^2\), where \(a = 4x - 3y\)
• \(64\) is \(b^2\), where \(b = 8\)

Hence, we have the form \((a^2 - b^2)\), which can be factored as \((a + b)(a - b)\). In our specific context, substituting \(a = 4x - 3y\) and \(b = 8\), we get: \((4x - 3y + 8)(4x - 3y - 8)\).

The concept of grouping terms is essential in factoring complex algebraic expressions. The idea is to reformat the expression in a way that highlights recognizable patterns. For the exercise at hand, we start with the quadratic expression \(16x^2 - 24xy + 9y^2 - 64\).

We observe and group terms as follows: \((16x^2 - 24xy + 9y^2) - 64\).

• The terms \(16x^2 - 24xy + 9y^2\) form a recognizable pattern of a perfect square trinomial.
• The constant \(-64\) stands alone, aiding in subsequent factoring.

By grouping the terms effectively, we can then factor the perfect square trinomial as \((4x - 3y)^2\), bringing the expression into a simpler form. This strategic grouping makes it easier to apply other factoring techniques, such as the difference of squares approach.

These steps illustrate the importance of recognizing patterns and strategically grouping terms to simplify and factor complex quadratic expressions.