Problem 95 - Elementary Algebra

Question

Use a pattern to multiply \((6 x+7)(6 x-7)\).

Step-by-Step Solution

Answer

36x^2 - 49

Step 1: Recognize the Pattern

Identify that the expression \( (6x+7)(6x-7) \) follows the difference of squares formula: \( (a+b)(a-b) = a^2 - b^2 \). Here, \( a = 6x \) and \( b = 7 \).

Step 2: Apply the Formula

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

Step 3: Simplify Each Term

Calculate \( (6x)^2 \) to get \( 36x^2 \), and \ 7^2 \ to get \ 49 \. This gives: \[ (6x+7)(6x-7) = 36x^2 - 49 \].

Key Concepts

Algebraic Patterns

Recognizing algebraic patterns is a powerful skill in algebra. One common pattern is the difference of squares. This pattern follows the formula: \[ (a + b)(a - b) = a^2 - b^2 \]. By identifying this pattern correctly, you can simplify many algebraic expressions with ease. In our example, \[ (6x + 7)(6x - 7) \] fits this pattern perfectly, where \[ a = 6x \] and \[ b = 7 \]. This knowledge simplifies multiplication and saves time.

Multiplication of Polynomials

Multiplication of polynomials might look tricky at first, but identifying common patterns can make it much simpler. For our exercise, we used the difference of squares formula to multiply \[ (6x + 7)(6x - 7) \]. This approach helps in avoiding lengthy multiplication of terms. Instead of multiplying each term individually, you immediately use the formula \ufffd \[ (a + b)(a - b) = a^2 - b^2 \]. By substituting \[ a = 6x \] and \[ b = 7 \], and then squaring these values, you directly get \[ 36x^2 - 49 \]. Practice spotting patterns to multiply polynomials quickly and accurately.

Simplification of Expressions

Simplifying algebraic expressions is an essential part of solving mathematical problems. Taking \[ (6x + 7)(6x - 7) = 36x^2 - 49 \] as an example, you can see that identifying and applying the appropriate algebraic pattern makes the simplification straightforward. Breaking the problem down: first, you recognize the pattern (difference of squares), then you apply the formula, and finally you compute the simplified terms \[ 36x^2 - 49 \]. Simplifying expressions helps in better understanding the underlying algebraic structures and solving equations efficiently.