Problem 59 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$(2 x+y-3)(2 x+y+3)$$

Step-by-Step Solution

Answer

The expression simplifies to \(4x^2 + 4xy + y^2 - 9\).

Step 1: Identify the Expression Type

The given expression \((2x + y - 3)(2x + y + 3)\) is a product of conjugates, which can be expanded using the formula \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 2x + y\) and \(b = 3\).

Step 2: Apply the Conjugate Product Formula

Using the formula \((a - b)(a + b) = a^2 - b^2\), substitute \(a = 2x + y\) and \(b = 3\). This results in \[(2x + y)^2 - 3^2\]

Step 3: Expand \((2x + y)^2\)

Expand \((2x + y)^2\) using the binomial expansion formula: \[(2x)^2 + 2(2x)(y) + (y)^2 = 4x^2 + 4xy + y^2\]

Step 4: Simplify \(3^2\)

Calculate \(3^2\), which is \(9\).

Step 5: Substitute and Simplify

Substitute the expanded terms back into the expression: \[(4x^2 + 4xy + y^2) - 9\]. Simplify this to \[(4x^2 + 4xy + y^2 - 9)\].

Key Concepts

Understanding Conjugate Pairs

In algebra, conjugate pairs are expressions structured in the form

• \((a + b)\) and \((a - b)\).

When these pairs are multiplied, the product is formed using the special identity

• \((a + b)(a - b) = a^2 - b^2\),

which simplifies the multiplication process significantly.

In the context of our exercise, the conjugate pairs are

• \((2x + y - 3)\) and \((2x + y + 3)\).

The terms \((2x + y)\) are considered as \(a\), while \(3\) is \(b\). By multiplying these conjugates using the formula, you can quickly get to

• \((2x + y)^2 - 3^2)\).

Recognizing conjugate pairs is a useful skill as it simplifies complex expressions into manageable forms.

Applying Binomial Expansion

Binomial expansion applies when you multiply terms raised to powers. It follows the formula:

• \( (a + b)^2 = a^2 + 2ab + b^2 \).

In this problem, we expanded the binomial

• \((2x + y)^2\) using this formula:

• First, calculate the square of each term: \((2x)^2 = 4x^2\) and \(y^2 = y^2\).
• Next, multiply \(2(2x)(y)\), which gives you \(4xy\).
• Combine these results:

\[4x^2 + 4xy + y^2\].

This expansion converts a binomial squared expression into a trinomial, easing complex calculations by revealing all its component terms.

Simplification of Expressions

Simplifying expressions involves reducing them to their simplest form. This process helps remove

• like terms, arithmetic simplification, and the application of algebraic identities.

After expanding the conjugate pair formula, we arrived at an expression

• \( (4x^2 + 4xy + y^2) - 9 \).

In this step, perform any arithmetic, such as simplifying \(3^2\) to \(9\), and substituting it back:

• \(4x^2 + 4xy + y^2 - 9\).

This is the final form, where no further simplification is possible in terms of combining like terms or further factorization.

Simplifying properly ensures the expression is clearer and more efficient to work with in further calculations or algebraic manipulations.