Problem 1 Melissa said that \((a+3)^{2}=a^... [FREE SOLUTION]

Chapter 1: Problem 1

Melissa said that \((a+3)^{2}=a^{2}+9 .\) Do you agree with Melissa? Justify your answer.

Short Answer

Expert verified

No, Melissa's expansion is incorrect; she missed the term \(6a\) in \((a+3)^{2} = a^2 + 6a + 9\).

Step by step solution

Understand the Mistake by Expanding the Expression

Melissa recommends the equality \((a+3)^{2}=a^{2}+9\). To check this, let's expand \((a+3)^{2}\). Using the formula for the square of a binomial, we have the expression \((a+b)^2 = a^2 + 2ab + b^2\). Applying this to \((a+3)^2\), we get \(a^2 + 2 \cdot a \cdot 3 + 3^2\).

Calculate Each Term

Let's calculate each of the terms in the expanded expression from Step 1. The first term is \(a^2\). The second term is \(2 \cdot a \cdot 3 = 6a\). The third term is \(3^2 = 9\).

Write the Correct Expanded Expression

Combine the terms calculated in Step 2 to get the fully expanded expression. Therefore, \((a+3)^{2} = a^2 + 6a + 9\). Notice the additional term \(6a\) that Melissa overlooked.

Comparison

Compare both expressions. Melissa thought \((a+3)^{2} = a^2 + 9\). However, the correct expression is \((a+3)^{2} = a^2 + 6a + 9\). The missing term \(6a\) causes Melissa's statement to be incorrect.

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Key Concepts

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

Binomial Expansion

The binomial expansion is a powerful tool in algebra that helps us simplify and expand expressions like

\((a+b)^n\).

In Melissa's case, we're looking at

\((a+3)^2\),
which is a specific scenario of binomial expansion where
the exponent is 2.

The general formula for expanding a binomial squared is

\((a+b)^2 = a^2 + 2ab + b^2\).

Each binomial expression

consists of two terms,
and the square involves three terms in the expanded form.

By applying this formula, we identify the three critical terms as

\(a^2\),
\(2ab\),
and \(b^2\).

This crucial middle term,

\(2ab\),
often gets mistakenly left out, resulting in false expansions
such as the one Melissa made.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is 2.
The general form is typically

\(ax^2 + bx + c\),

where

\(a\),
\(b\),
and \(c\)
are constants.

In expanded binomial expressions such as

\((a+3)^2\),
the result is always
a quadratic expression since it contains the \(a^2\) term.

The key to understanding quadratic expressions involves recognizing

these three parts: the square term, the linear term,
and the constant term.

In Melissa's error, she skipped the linear term,

\(6a\),
which completes the quadratic expression.

Understanding how to correctly form these expressions
helps avoid such mistakes.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations.
They are the backbone of algebra and come in various forms including binomials,
as seen in the original exercise:

\((a+3)\).

Algebraic expressions can be simplified or expanded depending on the requirement.
When we expand

\((a+3)^2\),
we鈥檙e taking an algebraic expression and expressing it as the sum of simpler terms.

The task involves recognizing how variables interact with constants through operations
such as addition and multiplication.

This understanding is crucial in expanding expressions correctly and ensuring
that no terms are mistakenly omitted, as Melissa did.

Mastery of algebraic expressions provides a strong foundation
for further mathematical studies and problem-solving.

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91影视

Melissa said that \((a+3)^{2}=a^{2}+9 .\) Do you agree with Melissa? Justify your answer.

Short Answer

Step by step solution

Understand the Mistake by Expanding the Expression

Calculate Each Term

Write the Correct Expanded Expression

Comparison

Key Concepts

Binomial Expansion

Quadratic Expressions

Algebraic Expressions

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