Problem 150 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \((y+2)^{2}-4\)

Step-by-Step Solution

Answer

The expression factors to \(y(y+4)\).

Step 1: Identify the Form

The expression \((y+2)^2 - 4\) can be seen as a difference of squares. Recall that a difference of squares is in the form \(a^2 - b^2\), which factors to \((a-b)(a+b)\). Here, \(a = (y+2)\) and \(b = 2\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\), we can factor the expression: \((y+2) - 2)((y+2) + 2)\).

Step 3: Simplify Each Factor

Simplify the terms in each factor. \((y+2) - 2 = y\) and \((y+2) + 2 = y+4\).

Step 4: Write the Final Factored Form

The completely factored expression is \(y(y+4)\).

Key Concepts

Difference of Squares

The difference of squares is an important algebraic identity that simplifies expressions of the form \(a^2 - b^2\). This identity states that \(a^2 - b^2 = (a-b)(a+b)\). It's one of the most straightforward factoring identities and very useful in algebraic manipulation.

When you recognize an expression as a difference of squares, it means you have two perfect squares being subtracted. This method was perfectly applied in this exercise since the expression \((y+2)^2-4\) can be seen as \((y+2)^2-(2)^2\).

Here's a brief checklist to help identify the difference of squares in future expressions:

• Are both terms perfect squares?
• Is there a subtraction (")-") sign between the two squares?
• Can it be rewritten in the form \(a^2-b^2\)?

Recognizing this pattern will make solving related problems much easier.

Quadratic Expressions

Quadratic expressions, like \((y+2)^2-4\), frequently appear in algebra. They generally take the form \(ax^2 + bx + c\). However, this particular exercise is a bit unique as it involves a quadratic disguised in a difference of squares.

This expression can initially be seen as a quadratic, particularly by noticing \((y+2)\) is squared. Here, \(y+2\) acts as a single unit or term, representing \(a\) in the standard form \(a^2 - b^2\).

Understanding that any binomial squared \((a+b)^2\) expands to \(a^2 + 2ab + b^2\) is vital in spotting quadratic forms, especially under a difference of squares setup. Recognizing these patterns will assist in tackling more complex algebraic expressions.

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle. It's about changing the form of an equation or expression to make it easier to work with or understand. In this exercise, we manipulated the expression \((y+2)^2-4\) using the difference of squares identity to simplify it.

We started by identifying the appropriate algebraic pattern (difference of squares), then used this to break down the expression into simpler components. Manipulating a quadratic expression required recognizing that \((y+2)\) needed to be treated as a whole.

To master algebraic manipulation, follow these steps:

• Identify patterns or identities that fit the expression.
• Rewrite the expression using these identities.
• Simultaneously simplify each step patiently.

With practice, these steps will become second nature, making algebra much more intuitive.