Problem 20 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ b^{2}-48 $$

Step-by-Step Solution

Answer

The factored form of the expression \(b^2 - 48\) is \((b+2\sqrt{3})(b-2\sqrt{3})\).

Step 1: Identify the squares in the expression

The expression given is \(b^2 - 48\). Here, the first term \(b^2\) is obviously a perfect square, with \(b\) being the square root. The second term, 48, is not a perfect square. But notice that 48 equals \(4^2 \times 3\), which means that it can be expressed as \((2\sqrt{3})^2\). So, the given expression \(b^2 - 48\) can be rewritten as \(b^2 - (2\sqrt{3})^2\).

Step 2: Apply the difference of squares rule

According to the rule \(a^2 - b^2 = (a+b)(a-b)\), an expression of the form \(b^2 - (2\sqrt{3})^2\) can be factored as \((b+2\sqrt{3})(b-2\sqrt{3})\).

Step 3: Write the final answer

The factored form of the expression \(b^2 - 48\) is \((b+2\sqrt{3})(b-2\sqrt{3})\).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic expression that arises when you have two squares subtracted from each other. This form looks like this: \(a^2 - b^2\). It's a powerful concept in factoring because there is a straightforward formula for it. The difference of squares formula is:

• \(a^2 - b^2 = (a + b)(a - b)\)

This simplifies the process of breaking down an expression into simpler parts, known as factors. In our original exercise where we have \(b^2 - 48\), although 48 is not a perfect square itself, we reconstructed it as \((2\sqrt{3})^2\) to apply this rule effectively. This way, you'll always find two products of sums and differences when using the difference of squares formula. It's particularly useful when simplifying expressions or solving equations.

Perfect Squares

A perfect square is a number or expression obtained by squaring a whole number or an equivalent algebraic term. For example, \(4\) and \(9\) are perfect squares as they are \(2^2\) and \(3^2\), respectively. Similarly, \(b^2\) is a perfect square since it is the square of \(b\).
To recognize perfect squares within algebraic expressions, look for terms that can be expressed as a number raised to the power of two. In our understanding of the expression \(b^2 - 48\), \(b^2\) is identified as a perfect square. Identifying perfect squares helps us to conveniently reconfigure them for different algebraic manipulations, such as in the difference of squares method we applied here. Remember, even if a term is not directly a perfect square like \(48\), you might be able to express it in terms of perfect squares, as shown when rewritten to \((2\sqrt{3})^2\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations (like addition, subtraction, multiplication, etc.). These are foundational in understanding algebra as they represent real-world problems in mathematical terms.
Factors come into play when breaking down expressions into simpler or more manageable pieces for solving or transformation purposes. Our original problem, \(b^2 - 48\), was an algebraic expression that we needed to factor. By understanding and using the key concepts of difference and perfect squares, we were able to express \(b^2 - 48\) as \((b + 2\sqrt{3})(b - 2\sqrt{3})\).
Recognizing the form and components of algebraic expressions is crucial for solving equations, factoring polynomials, and learning more advanced mathematics. Mastering these basics equips you with tools to handle and simplify any parts of algebra.