Problem 96 - Elementary Algebra

Question

Use the distributive property to multiply \((6 x+7)(6 x-7)\).

Step-by-Step Solution

Answer

The result is \ 36x^2 - 49. \

Step 1: Identify the Formula

Recognize that the expression \( (6x+7)(6x-7) \) is in the form of \( (a + b)(a - b) \). According to the difference of squares formula, \[(a + b)(a - b) = a^2 - b^2.\]

Step 2: Assign Values to Variables

Assign the values: \( a = 6x \) and \( b = 7 \).

Step 3: Apply the Difference of Squares Formula

Substitute \( a \) and \( b \) into the difference of squares formula: \[(6x + 7)(6x - 7) = (6x)^2 - 7^2.\]

Step 4: Simplify Each Term

Simplify the terms separately: \[ (6x)^2 = 36x^2 \] and \[ 7^2 = 49. \]

Step 5: Combine the Results

Combine the simplified terms to get the final result: \[(6x)^2 - 7^2 = 36x^2 - 49. \]

Key Concepts

Difference of Squares Formula

The Difference of Squares Formula is a very handy tool in algebra. It simplifies expressions of the form \(a^2 - b^2\). The formula is: \[(a + b)(a - b) = a^2 - b^2.\] This means that if you multiply together a binomial with the same binomial where the only difference is the sign (positive and negative), you get the difference of two squares. In the exercise you had, the original expression \((6x+7)(6x-7)\) follows this pattern precisely. The value for \(a\) is \(6x\) and the value for \(b\) is \(7\). You substitute those values and simply apply the formula, leading to a simpler form.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra. To simplify means to make an expression as straightforward as possible while maintaining its equivalence. The key steps include:

• Identifying like terms: These are terms that have the same variables raised to the same power.
• Combining like terms: Add or subtract the coefficients of like terms.
• Using algebraic formulas: These are shortcuts that apply to specific forms of expressions such as the difference of squares.

In the provided exercise, after applying the difference of squares formula, we simplified: \( (6x)^2 - 7^2\) to get \(36x^2 - 49\). This is the most simplified form because no further reduction or combining of like terms is possible.

Multiplying Binomials

Multiplying binomials is a common task in algebra. When you multiply two binomials, you apply the distributive property. Here's the pattern:

• Use the FOIL method (First, Outer, Inner, Last) if both terms are binomials like \((a+b)(c+d)\).
• Distribute each term in the first binomial to each term in the second binomial.

However, if the binomials fit certain special patterns like the difference of squares, use the appropriate formula directly as it simplifies the process greatly. For example, the term provided, \((6x + 7)(6x - 7)\), did fit the special pattern and using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) made the expression simpler and quicker to solve. So, understanding these patterns and when to use them can save a lot of time and effort in algebra.