Problem 46 - Elementary Algebra

Question

For the following problems, select a number (or numbers) to show that \((a+b)^{2}\) is not generally equal to \(a^{2}+b^{2}\).

Step-by-Step Solution

Answer

Question: Demonstrate with an example that \((a+b)^2\) is not generally equal to \(a^2+b^2\). Answer: Consider the numbers a=2 and b=3. Calculating (a+b)^2, we have (2+3)^2 = 5^2 = 25. For a^2+b^2, we get 2^2 + 3^2 = 4 + 9 = 13. Since 25 ≠ 13, we have shown that \((a+b)^2\) is not generally equal to \(a^2+b^2\).

Step 1: Select a pair of numbers to test

Let's choose the numbers a=2 and b=3.

Step 2: Plug the numbers into the first expression \((a+b)^2\)

Plugging a=2 and b=3 into \((a+b)^2\), we get: \((2+3)^2\)

Step 3: Calculate the value of the expression\((a+b)^2\)

Evaluating the expression, we have: \((2+3)^2 = 5^2 = 25\)

Step 4: Plug the numbers into the second expression \(a^2+b^2\)

Now, plugging a=2 and b=3 into \(a^2+b^2\): \(2^2+3^2\)

Step 5: Calculate the value of the expression\(a^2+b^2\)

Evaluating the expression, we have: \(2^2+3^2 = 4+9 = 13\)

Step 6: Compare the values of both expressions

Our results show that: \((a+b)^2 = 25 \neq 13 = a^2+b^2\) Since the values of both expressions are not equal, we have shown that \((a+b)^2\) is not generally equal to \(a^2+b^2\) by providing a counterexample.

Key Concepts

Binomial Expansion

The binomial expansion is a way of expressing powers of binomials. A binomial is simply a two-term expression, such as \((a + b)\). When expanded, \((a + b)^2\) becomes \(a^2 + 2ab + b^2\). This formula is derived from multiplying \((a + b)\) by itself, which is why it includes the \(2ab\) term.

It's crucial to understand that this extra \(2ab\) term prevents \((a + b)^2\) from being just \(a^2 + b^2\). The binomial expansion shows us why the product changes rather than simply squaring two separate terms independently.

• Always remember: The formula for \((a+b)^2\) is \(a^2 + 2ab + b^2\).
• The \(2ab\) is what differentiates the binomial expansion from just adding squared terms \(a^2\) and \(b^2\).

Understanding binomial expansion helps us accurately work with expressions where terms are added and then squared.

Counterexample

A counterexample is a helpful tool in mathematics for demonstrating that a certain statement is not always true. In the exercise, we used a counterexample to show that \((a+b)^2\) is not equal to \(a^2 + b^2\).

By choosing specific numbers like \(a=2\) and \(b=3\), we proved that the two expressions yield different results: \((5^2 = 25)\) as opposed to \(4 + 9 = 13\).

Finding a single instance where the claim doesn’t hold true is powerful because it disproves the universal validity of the statement.

• A counterexample highlights the exception, showcasing why a statement cannot be accepted as true in all cases.
• Just a single disproving example suffices to show the claim is not generally valid across all numbers.

Using counterexamples keeps us honest in mathematics, prompting us to think critically and check assumptions.

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree two, meaning their highest exponent on the variable is two. The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

When we expand \((a + b)^2\), we're dealing with a specific type of quadratic expression, which becomes \(a^2 + 2ab + b^2\).

The \(2ab\) term is key here as it confirms that we're indeed forming a quadratic.

• Quadratic expressions often appear in various areas of algebra, making them important to understand deeply.
• Recognizing these expressions helps in solving equations, graphing parabolas, and analyzing functions.

Mastering quadratic expressions is essential for further study in mathematics, as they form the foundation for more complex algebraic operations and concepts.