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Chapter 0: Problem 79

Simplify each expression. $$ (6 x+5)(6 x-5)-(6 x+5)^{2} $$

Short Answer

Expert verified
-60x - 50

Step by step solution

01

Apply the Difference of Squares Formula

Start by expanding the term oindent\( (6x + 5)(6x - 5) = (6x)^2 - 5^2 = 36x^2 - 25 \). This is a direct application of the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\)
02

Expand the Second Term

Now, expand the binomial square \( (6x + 5)^2 \):oindent\( (6x+5)^2 = (6x)^2 + 2(6x)(5) + 5^2 = 36x^2 + 60x + 25 \)
03

Substitute and Simplify

Substitute the expanded terms into the original expression:oindent\( (6x + 5)(6x - 5) - (6x + 5)^2 = 36x^2 - 25 - (36x^2 + 60x + 25) \). Now combine and simplify the expression:oindent\( 36x^2 - 25 - 36x^2 - 60x - 25 = -60x - 50 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a special algebraic formula used to simplify expressions of the form \(a^2 - b^2\). This pattern arises when you multiply a binomial by its conjugate. A conjugate is the same binomial but with the opposite sign between the terms. For instance, in \( (6x+5)(6x-5) \), \( (6x+5) \) and \( (6x-5) \) are conjugates.
The formula is given by \[(a + b)(a - b) = a^2 - b^2 \].
Let's apply this to the problem:
\[(6x+5)(6x-5)\rightarrow(6x)^2 - 5^2 = 36x^2 - 25\]
Knowing the difference of squares helps you quickly simplify expressions involving multiplied conjugates.
Binomial Expansion
Binomial expansion refers to expanding an expression that is raised to a power, such as \( (a+b)^2 \). For our exercise, we need to expand \( (6x+5)^2 \). The formula for the square of a binomial is given by:
\[(a + b)^2 = a^2 + 2ab + b^2 \]
Applying this to our expression, we get:
\[(6x+5)^2 \rightarrow (6x)^2+2(6x)(5)+5^2\]
Simplifying this gives us:
\[36x^2 + 60x + 25\]
Understanding binomial expansion is crucial as it allows you to convert a squared binomial into a sum of terms, which can then be used in further algebraic simplification.
Simplification Steps
Simplifying algebraic expressions involves following specific steps to combine like terms and reduce the expression to its simplest form. Here鈥檚 a detailed breakdown of the steps we used in the original exercise:
  • First, apply the difference of squares formula to \( (6x+5)(6x-5) \) and expand to get \ 36x^2 - 25 \.
  • Next, expand the second term, \( (6x+5)^2\), using the binomial expansion formula: \ 36x^2 + 60x + 25 \.
  • Substitute these expanded expressions back into the original: \ 36x^2 - 25 - (36x^2 + 60x + 25) \.
  • Combine like terms and simplify: \ 36x^2 - 25 - 36x^2 - 60x - 25 = -60x - 50 \.
Breaking down the steps in this manner makes it easier to follow and understand the simplifying process.

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