Problem 96 - Algebra A Combined Function

Question

Evaluate each of the following. a. \((2+3)^{2} ; 2^{2}+3^{2}\) b. \((8+10)^{2} ; 8^{2}+10^{2}\) c. Does \((a+b)^{2}=a^{2}+b^{2}\) no matter what the values of \(a\) and \(b\) are? Why or why not?

Step-by-Step Solution

Answer

No, \((a+b)^2\) is not equal to \(a^2 + b^2\) because \((a+b)^2\) includes an extra \(2ab\) term.

Step 1: Evaluate Part (a)

First, calculate \((2+3)^2\). Simplify the expression inside the parentheses: \(2+3=5\). Then square the result: \(5^2 = 25\). Next, separately, calculate \(2^2 + 3^2\). \(2^2 = 4\) and \(3^2 = 9\). Add these two results: \(4 + 9 = 13\).

Step 2: Compare Part (a)

In step 1, we found that \((2+3)^2 = 25\) and \(2^2 + 3^2 = 13\). Notice that \(25 eq 13\), indicating that the expressions are not equal.

Step 3: Evaluate Part (b)

First, calculate \((8+10)^2\). Simplify the expression inside the parentheses: \(8+10=18\). Then square the result: \(18^2 = 324\). Next, separately, calculate \(8^2 + 10^2\). \(8^2 = 64\) and \(10^2 = 100\). Add these two results: \(64 + 100 = 164\).

Step 4: Compare Part (b)

In step 3, we found that \((8+10)^2 = 324\) and \(8^2 + 10^2 = 164\). Notice that \(324 eq 164\), indicating again that the expressions are not equal.

Step 5: Concluding Part (c)

From the calculations in parts (a) and (b), we observed that \((a+b)^2\) does not equal \(a^2 + b^2\). This is because \((a+b)^2\) expands to \(a^2 + 2ab + b^2\), which includes an additional \(2ab\) term. Therefore, \((a+b)^2 = a^2 + b^2\) is not true for any values of \(a\) and \(b\).

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree two, featuring a variable raised to the power of two. They often resemble the form: \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. Quadratic expressions can represent various practical situations, like projectile motion or area calculations.
To evaluate or manipulate these expressions, it's crucial to understand how each component interacts. For instance, the coefficient \(a\) impacts the parabola's width and direction, whereas \(b\) and \(c\) adjust its placement on the graph.
Understanding quadratic expressions is essential when working with quadratic equations or performing algebraic expansions. They lay the foundation for more advanced topics in math, making them a valuable tool for students.

Algebraic Expansion

Algebraic expansion involves multiplying out expressions or brackets and simplifying them into their polynomial form. The binomial theorem is particularly useful when expanding expressions of the form \((a+b)^n\).
One common scenario involves squaring a binomial, transforming \((a+b)^2\) into \(a^2 + 2ab + b^2\). This process highlights that when expanding, it is essential to remember the middle term s\(2ab\), which often causes confusion if omitted.

• Multiplying terms directly involves each term within a bracket multiplied by every term in another.
• The foil method can help, which stands for First, Outer, Inner, Last, indicating the order of multiplication.

Understanding algebraic expansion is fundamental because it allows you to simplify complex algebraic expressions, making problem-solving more efficient and avoiding common mistakes.

Square of a Binomial

Squaring a binomial is a special case of algebraic expansion where you take the binomial expression \((a+b)\) and square it, symbolized as \((a+b)^2\).
It's essential to recognize that \((a+b)^2\) is not simply \(a^2 + b^2\). Instead, it expands to \(a^2 + 2ab + b^2\). The middle term, \(2ab\), signifies the interaction between \(a\) and \(b\), and its presence leads to the correct full expansion.
This pattern is consistent whenever squaring any binomial expressions.

• Understand that this happens due to the distributive property, where each term in the binomial is multiplied by each other.
• Always verify the presence of all components—\(a^2\), \(2ab\), and \(b^2\)—to ensure accuracy.

Recognizing and applying the square of a binomial helps in correctly solving and simplifying algebraic problems, ensuring clarity and precision in your work.