Problem 52 - Algebra A Combined Function

Question

Multiply. $$ \left(10 x+\frac{2}{7}\right)\left(10 x-\frac{2}{7}\right) $$

Step-by-Step Solution

Answer

The product is \(100x^2 - \frac{4}{49}\).

Step 1: Identify the expression format

The expression is of the form \((a+b)(a-b)\), which is a difference of squares.

Step 2: Apply the difference of squares formula

The difference of squares formula states that \((a+b)(a-b) = a^2 - b^2\). In our expression, \(a = 10x\) and \(b = \frac{2}{7}\).

Step 3: Square the terms

Calculate \(a^2 = (10x)^2 = 100x^2\) and \(b^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49}\).

Step 4: Substitute the squared terms into the formula

Replace \(a^2\) and \(b^2\) in the formula: \(100x^2 - \frac{4}{49}\).

Step 5: Final result

The product of \((10x + \frac{2}{7})(10x - \frac{2}{7})\) simplifies to \(100x^2 - \frac{4}{49}\).

Key Concepts

Multiplication of Binomials

When multiplying binomials, you're working with two algebraic expressions that are added or subtracted from each other. Binomials have the form \((m + n)(m - n)\), where \(m\) and \(n\) are terms that may include numbers, variables, or both.
Understanding this concept is crucial in algebra as it often appears in polynomial expressions and advanced mathematics.

To multiply binomials like these, there are methods you can use:

• FOIL Method: This stands for First, Outer, Inner, Last, representing the order of multiplying terms.
• Using identities: In this context, the formula \((a+b)(a-b) = a^2 - b^2\) directly applies, simplifying the multiplication process significantly.

Recognizing when to apply such identities allows you to streamline solving problems. By identifiying patterns like the difference of squares, you can quickly come to a result, as seen in the provided original solution.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They consist of terms, coefficients, and variables arranged in meaningful ways.
In the expression \((10x + \frac{2}{7})(10x - \frac{2}{7})\), we have two binomials.

An algebraic expression can feature:

• Variables: Symbols like \(x\), representing unknown quantities.
• Coefficients: Numbers keeping the variable's company, such as \(10\) in \(10x\).
• Constants: Fixed values, like \(\frac{2}{7}\) in our example.

These components work together allowing us to describe mathematical relationships. When using these expressions, operations like addition, subtraction, and multiplication help us manipulate them to find solutions. Recognizing the structure of an algebraic expression like distinguishing terms or identifying coefficients is essential in simplifying and solving equations.

Squared Terms

Squared terms are fundamental in algebra, appearing in various forms of equations and expressions.
When you square a term, you multiply it by itself. For instance, in \( (10x)^2 \), you multiply \(10x\) by \(10x\).

The rules when handling squared terms include:

• Squaring variables: For any variable \(x\), \(x^2\) denotes \(x\) multiplied by itself.
• Squaring coefficients: Numbers like \(10\) are squared separately from the variable, so \(10^2 = 100\).
• Overall computation: Both results are combined. Therefore, \((10x)^2\) equals \(100x^2\).

When dealing with expressions like the difference of squares, recognizing the structure \(a^2 - b^2\) helps streamline the process.
Thus, computing squared terms accurately is critical for grasping algebraic concepts and solving related problems.