Problem 1 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(a^2 + 2 \cdot a \cdot 3 + 3^2\).

Step 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 \(3^2 = 9\).

Step 3: 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.

Step 4: 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.

Key Concepts

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’re 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.