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Chapter 6: Problem 231

Explain why \(n^{2}+25 \neq(n+5)^{2}\). Use algebra, words, or pictures.

Short Answer

Expert verified
Expanding \((n+5)^2\) gives \(n^2 + 10n + 25\), which shows an extra \(10n\) term.

Step by step solution

01

Understand the Equation

We need to explain why the equation \(n^{2} + 25 eq (n+5)^{2}\).
02

Expand \((n+5)^{2}\)

Begin by expanding the right side of the equation: \[(n+5)^{2} = (n+5)(n+5).Use the distributive property to expand it: (n+5)(n+5) = n*n + n*5 + 5*n + 5*5 = n^{2} + 10n + 25.\]
03

Compare Both Sides

Now compare both sides of the original equation:\[n^{2} + 25\] and \[n^{2} + 10n + 25.\]We can see clearly that \(n^{2} + 25\) does not include the \(10n\) term present in \(n^{2} + 10n + 25\).
04

Conclude the Difference

Hence, due to the extra \(10n\) term in \(n^{2} + 10n + 25\), we can conclude that:\[n^{2} + 25 eq (n+5)^{2}.\]

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Key Concepts

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

Expanding Binomials
Expanding binomials might sound tricky, but it's actually very straightforward when you understand the steps. Let's start with the binomial \((n+5)\). When we square this binomial, we get \((n+5)^2\), which means \((n+5)*(n+5)\). To expand this, we need to use the distributive property to multiply each term in the first binomial by each term in the second binomial. \((n+5)(n+5)\) becomes \((n*n + n*5 + 5*n + 5*5)\), which simplifies to \((n^2 + 10n + 25)\). Breaking it down like this helps us expand binomials correctly and avoid mistakes.
Distributive Property
Understanding the distributive property is essential for expanding binomials and simplifying algebraic expressions. The distributive property allows us to multiply a sum by distributing the multiplication over each addend inside the parenthesis. For instance, in our binomial expansion \((n+5)(n+5)\), we use the distributive property to distribute \(n\) and \(5\) to both terms. We get four products: \(n*n, n*5, 5*n, \text{and} 5*5\). Adding these up, we get \((n^2 + 10n + 25)\). This property ensures that we account for every possible multiplication, maintaining the equality throughout the process.
Algebraic Expressions
Algebraic expressions like \((n+5)^2\) and \((n^2 + 25)\) are made up of numbers, variables, and operators. When we say \((n+5)^2\), we refer to an expression that follows specific algebraic rules to simplify or expand. Both expressions look similar but contain different terms. Our task is to show the difference. After expanding \((n+5)^2\) to \((n^2 + 10n + 25)\), we can compare it side-by-side with \((n^2 + 25)\). We quickly see \((n^2 + 25)\) lacks the \(10n\) term found in \((n^2 + 10n + 25)\). So, algebraic expressions must be carefully analyzed, expanded, and compared to ensure we correctly represent the given mathematical problems.

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