Problem 92 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (a-3)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the algebraic expression \((a-3)^2\) is \(a^2 - 6a + 9\).

Step 1: Identify the expression

We are given the algebraic expression \((a-3)^2\) and need to simplify it.

Step 2: Expand the expression using the distributive property (FOIL method)

To expand the expression, we will multiply \((a-3)\) by itself: \((a-3)(a-3)\). Using the distributive property, or the FOIL method (First, Outer, Inner, Last), we get: $$ (a-3)(a-3) = a^2 - 3a - 3a + 9 $$

Step 3: Combine like terms

Now, we will combine the like terms, which in this case are the terms \(-3a\) and \(-3a\): $$ a^2 - 3a - 3a + 9 = a^2 - 6a + 9 $$

Step 4: Write the final simplified expression

After expanding and combining like terms, the simplified expression is: $$ (a-3)^2 = a^2 - 6a + 9 $$

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators (like addition, subtraction, multiplication, or division). These expressions can represent real-life quantities and are used extensively in equations and functions. Understanding them is key to solving many problems in algebra.
A typical algebraic expression can be something like \(3x + 2\), where:

• "3" is the coefficient, meaning it tells how many times the variable is multiplied.
• "x" is the variable, which represents an unknown value that we solve for.
• "2" is the constant term, which stays the same regardless of the variable's value.

In the context of this exercise, \(a-3\)^2 is an algebraic expression and involves a variable "a" that you're simplifying. Mastering these basics can help you grasp more complex algebraic concepts.

Distributive Property

The distributive property is a key concept in simplifying algebraic expressions. It allows us to multiply a single term by each term within a set of parentheses. This method is especially useful when dealing with expressions like \(a-3\)\(a-3\).
To apply the distributive property effectively:

• Multiply each term in the first set of parentheses by every term in the second set.
• Combine the resulting products.

In our example, it's about expanding \(a-3\)^2\ by writing it as \(a-3\)(a-3)\ and distributing the terms. This results in separate products like \(a \cdot a, a \cdot (-3), -3 \cdot a,\) and \(-3 \cdot -3\), equating to \(a^2 - 3a - 3a + 9\). Thus, we can further simplify by using other methods.

Foiling Method

The FOIL method is a practical application of the distributive property for multiplying two binomials. It's an acronym which stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms.

• First: Multiply the first terms in both binomials, i.e., \(a \cdot a = a^2\).
• Outer: Multiply the outer terms, i.e., \(a \cdot -3 = -3a\).
• Inner: Multiply the inner terms, i.e., \(-3 \cdot a = -3a\).
• Last: Multiply the last terms in each binomial, i.e., \(-3 \cdot -3 = 9\).

This step results in the expanded expression \(a^2 - 3a - 3a + 9\). This method excels in simplifying expressions and equations with ease and accuracy.

Combining Like Terms

After expanding an algebraic expression, we simplify it by combining like terms. Like terms are terms whose variables and their exponents match. This process helps in making the expression more concise and easier to work with.
In the expression \(a^2 - 3a - 3a + 9\), the like terms are the ones containing the \(a\) variable:

• -3a and -3a, which add up to give -6a.

When you combine these terms, the expression becomes \(a^2 - 6a + 9\).
Combining like terms is a crucial step in algebra to ensure you have the most reduced and simplest version of your expression. This can be especially important in solving equations and understanding functions.