Problem 50 - Elementary and Intermediate Algebra

Question

Factor. $$ 49 d^{2}-16 $$

Step-by-Step Solution

Answer

The expression factors to \((7d - 4)(7d + 4)\).

Step 1: Recognizing a Difference of Squares

The expression given is \(49d^2 - 16\). This follows the pattern of \(a^2 - b^2\), which is known as a difference of squares. Here, \(a = 7d\) and \(b = 4\), since \((7d)^2 = 49d^2\) and \(4^2 = 16\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Substitute \(a = 7d\) and \(b = 4\) into this formula: \(a^2 - b^2 = (7d - 4)(7d + 4)\).

Step 3: Simplify the Expression

After applying the formula, we have \((7d - 4)(7d + 4)\). This is the fully factored form of the original expression \(49d^2 - 16\).

Key Concepts

Factoring

Factoring is like solving a puzzle where you break down complex expressions into simple multipliers. This skill is fundamental in algebra, and the difference of squares is one of its key techniques. Recognizing the pattern, such as \( a^2 - b^2 \), allows you to simplify expressions efficiently. In the given exercise, we identify the expression \( 49d^2 - 16 \) as a difference of squares. Here, both \( 49d^2 \) and \( 16 \) are perfect squares. Breaking them down, we get \( (7d)^2 \) and \( 4^2 \) respectively. Once identified, use the formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

Substitute the values, yielding \( (7d - 4)(7d + 4) \). Factoring like this not only simplifies expressions but plays a crucial role in solving equations later.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent specific values. They include terms like \( 49d^2 \) and \( -16 \) in this exercise. Understanding these expressions allows mathematicians to describe patterns and relationships efficiently.
Expressions can be simplified or factored. Factoring, in particular, breaks down an expression into a product of simpler expressions, making them easier to handle. In the context of differences of squares, recognizing square components is crucial for efficient factoring.
For example, identifying \( 49d^2 \) as \( (7d)^2 \) shows the expression's underlying structure. Once identified, apply algebraic techniques to simplify or manipulate expressions, laying the groundwork for advanced mathematical problem-solving.

Mathematics Education

Math education introduces students to various techniques crucial for higher-level mathematics, such as factoring. Mastering the difference of squares helps in simplifying complex expressions and solving quadratic equations. Educators typically focus on key patterns like
\( a^2 - b^2 \), urging students to recognize and apply these in different contexts. Such patterns can appear in multiple problem types, highlighting the importance of practice and familiarity.

• Deep discussions about the logic behind patterns enhance understanding.
• Hands-on practice reinforces learning and intuition for future problems.

Understanding foundational concepts through exercises like factoring enriches a student's mathematical journey. It builds confidence, promotes problem-solving skills, and sets the stage for more advanced topics like calculus and beyond.