Problem 57 - Algebra 2

Question

REASONING. Find a counterexample to the statement \(a^{2}+b^{2}=(a+b)^{2}\)

Step-by-Step Solution

Answer

The statement is false for \( a = 1 \) and \( b = 1 \).

Step 1: Analyze the Given Statement

The statement presents the equation \( a^2 + b^2 = (a+b)^2 \). We need to determine under which conditions this statement might not hold true. First, let's expand the right-hand side of the equation: \((a+b)^2 = a^2 + 2ab + b^2\). Thus, our original equation becomes \( a^2 + b^2 = a^2 + 2ab + b^2 \).

Step 2: Identify the False Condition

For the equation \( a^2 + b^2 = a^2 + 2ab + b^2 \) to be true, the term \( 2ab \) must equal zero because both sides are equal only if \( a^2 + b^2 = a^2 + 2ab + b^2 \) equates to the original form. Thus, we need to find values of \( a \) and \( b \) for which \( 2ab eq 0 \).

Step 3: Choose Values for Counterexample

Let's choose \( a = 1 \) and \( b = 1 \). Substituting these values into the original statement:- Left-hand side: \( a^2 + b^2 = 1^2 + 1^2 = 1 + 1 = 2 \).- Right-hand side: \( (a+b)^2 = (1+1)^2 = 2^2 = 4 \).

Step 4: Verify the Difference

Since \( 2 eq 4 \), the original statement \( a^2 + b^2 = (a+b)^2 \) is false for \( a = 1 \) and \( b = 1 \).Thus, \( a = 1 \) and \( b = 1 \) serves as a valid counterexample because both sides of the equation do not equal each other.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use one or more variables to describe relationships between quantities. These equations are vital tools for reasoning and problem solving. The objective often involves finding values for the variables that make the equation true. In our exercise, we see an equation:\[a^2 + b^2 = (a+b)^2\]Our task is to find a counterexample, which is a specific example that disproves the equation under certain conditions. We found that when expanding the equation, additional terms appear on one side, rendering the statement invalid when specific values for \(a\) and \(b\) are chosen. This illustrates the importance of correctly interpreting the terms of an equation, because an incorrect assumption can lead to flawed conclusions. This understanding is crucial when handling any equation to ensure the solutions are valid.

Expanding Expressions

Expanding expressions involves rewriting a mathematical statement to reveal its components, often by removing parentheses. In algebra, this skill is fundamental as it helps to simplify complex equations and identify mistakes in reasoning. For instance, when we expand the expression \[(a+b)^2\]we rewrite it as \[a^2 + 2ab + b^2\].Notice how the expanded form introduces the term \(2ab\), which is critical in determining the equation’s validity in the exercise.

• We expanded to compare both sides of the equation properly.
• This step revealed that additional terms can alter the balance of the statement.

Without expanding, one might incorrectly conclude that both expressions are equivalent. Thus, learning to expand expressions properly allows students to uncover the true components of a mathematical equation, leading to more accurate understanding and solutions.

Problem-Solving Strategies

Problem-solving strategies involve structured methods that aid in tackling mathematical problems effectively. Students use such strategies to simplify and systematically evaluate algebraic equations. Let's look at the steps used to verify the given statement in the exercise.First, we analyzed the equation\[a^2 + b^2 = (a+b)^2\]by expanding the right side to identify discrepancies - a crucial initial step.

• Expansion allowed us to spot the inconsistency \(2ab\) that wasn't present on the left side.
• Next, we hypothesized that \(a = 1\) and \(b = 1\) would help us verify our findings through substitution.
• Finally, we confirmed our hypothesis by comparing both sides and observing the difference.

These strategies ensure a comprehensive approach to problem solving, helping students not only to find counterexamples but also to deepen their understanding of how equations work.