Problem 110 - Algebra 2

Question

Factor each expression. $$ \frac{1}{4} b^{2}-4 $$

Step-by-Step Solution

Answer

The factored form of the expression \(\frac{1}{4} b^{2} - 4\) is \((\frac{1}{2}b - 2)(\frac{1}{2}b + 2)\)

Step 1: Identify the square expressions

Here, \(\frac{1}{4} b^{2}\) is a square of \(\frac{1}{2}b\) and \(4\) is a square of \(2\). So, our \(a\) is \(\frac{1}{2}b\) and \(b\) is \(2\)

Step 2: Apply the formula of the difference of squares

Now that we have identified \(a\) and \(b\), we can substitute them into the formula. Applying the formula we get: \((a-b)(a+b)\), or equivalently \((\frac{1}{2}b - 2)(\frac{1}{2}b + 2)\)

Step 3: Final expression

After plugging in the values, we obtain the final factored form of the given expression: \((\frac{1}{2}b - 2)(\frac{1}{2}b + 2)\)

Key Concepts

Difference of Squares

The difference of squares is a special pattern found in algebra where two perfect squares are subtracted from each other. This pattern follows the identity:

• \[a^2 - b^2 = (a - b)(a + b)\]

This means that if you can recognize an expression as a difference of squares, you can factor it quickly using this formula. In the exercise \[\frac{1}{4}b^2 - 4\], we identify \[\frac{1}{4}b^2\] as \((\frac{1}{2}b)^2\) and \[4\] as \((2)^2\). Recognizing these squares helps us directly apply the difference of squares formula, leading to the factors \((\frac{1}{2}b - 2)(\frac{1}{2}b + 2)\).The key is to look for two terms that are both perfect squares and connected by a subtraction, which signals the opportunity to use this straightforward factoring technique.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Expressions like \(\frac{1}{4}b^2 - 4\) include variables, such as \(b\), and constants, such as numbers that do not change. In algebra, expressions can often be rewritten in different forms for simplification or to solve equations.
Factoring expressions is a method of rewriting them to highlight particular properties or to make solving easier. In our example, the expression \(\frac{1}{4}b^2 - 4\) can be complex. But identifying it as a difference of squares allows us to simplify it significantly. The resulting factored form, \((\frac{1}{2}b - 2)(\frac{1}{2}b + 2)\), makes it more usable in calculations or further algebraic manipulations.Understanding how to manipulate these expressions is fundamental to mastering algebra.

Factoring Techniques

Factoring is the process of breaking down an expression into products of simpler expressions. It's a crucial skill in algebra because it simplifies expressions and is essential for solving equations. Here are some key factoring techniques:

• **Difference of Squares**: Recognized as \(a^2 - b^2 = (a - b)(a + b)\). Very useful when dealing with the subtraction of two squares.
• **Greatest Common Factor (GCF)**: The largest factor shared by all terms. By factoring out the GCF, you simplify the expression.
• **Trinomials**: Factoring expressions like \(ax^2 + bx + c\), requires techniques like splitting the middle term or using the quadratic formula.

To tackle the problem of factoring \(\frac{1}{4}b^2 - 4\), we specifically used the difference of squares technique. This illustrative example shows how recognizing specific patterns allows for efficient simplification.
Factoring is not just about mechanically applying techniques but understanding the underlying structure of expressions.