Problem 16 - Elementary Algebra

Question

For the following problems, find the products. $$ (a-6)^{2} $$

Step-by-Step Solution

Answer

Question: Find the product of the expression \((a-6)^2\). Answer: \((a-6)^2 = a^2 - 12a + 36\)

Step 1: Identify the binomial

The given expression is \((a-6)^2\), which represents the square of the binomial \((a-6)\). So, our binomial is: $$ (a - 6) $$

Step 2: Use the square of a binomial formula

Now, we'll use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). In our case, \(a\) is the variable \(a\), and \(b\) is the number \(6\). So, we'll substitute these values into the formula: $$ (a - 6)^2 = a^2 - 2(a)(6) + 6^2 $$

Step 3: Simplify the expression

Now, we need to simplify the expression. We'll calculate the products and combine like terms: $$ a^2 - 12a + 36 $$ So, the product of the given expression is: $$ (a - 6)^2 = a^2 - 12a + 36 $$

Key Concepts

Simplifying Algebraic Expressions

When dealing with algebra, simplifying expressions is a fundamental skill that helps make complex problems more manageable. Simplification often involves combining like terms, factoring, expanding products, and cancelling out terms when possible.

Combine Like Terms

Within an algebraic expression, terms that have the same variable part can be added or subtracted. For example, in the expression \(2x + 3x - x\), the terms are all alike because they share the same variable 'x', and we can simplify the expression to \(4x\).

Factor and Expand

Factoring involves expressing an expression as a product of its factors. This can make it easier to simplify or solve equations. Conversely, expanding involves the distributive property, commonly used when dealing with binomial products, such as squaring a binomial.

Use Algebraic Formulas

Recognizing and applying algebraic formulas such as the square of a binomial can greatly simplify the process of squaring these expressions. Instead of performing the multiplication manually, these formulas provide a quick and consistent method to arrive at the simplified expression.

Squaring Binomials

Squaring binomials is an essential operation within algebra that students frequently encounter. A binomial is an algebraic expression containing two terms. When squaring a binomial, you are essentially multiplying the binomial by itself.

Visualizing the Square

The process can be visualized as creating a square where each side represents the binomial, hence the term 'squaring'. When you calculate the area of the square, you multiply the length by the width, which corresponds to multiplying the binomial by itself.

Formula Application

The formula \( (a + b)^2 = a^2 + 2ab + b^2 \) provides a fast track to squaring any binomial. For the binomial \( a - 6 \) squared, applying the formula correctly is critical: \( (a - 6)^2 = a^2 - 2ab + b^2 \) reduces to \( a^2 - 12a + 36 \) after plugging in \( a \) and \( b \) (in this case, \( b = 6 \) ).

Algebraic Formulas

Algebraic formulas serve as shortcuts to solve problems without the need for complex calculations. There are a variety of these formulas, which include but are not limited to the square of a binomial, the difference of squares, and the quadratic formula.

Importance of Memorization

Having these formulas memorized allows students to recognize situations in which they can be applied, enabling quicker and more accurate problem-solving.

Applying the Formula

In our example, the formula for the square of a binomial was used to transform \( (a - 6)^2 \) into its simplified form \( a^2 - 12a + 36 \) seamlessly. Remember, the use of these formulas is not only a matter of application but also understanding when and how to use them appropriately in different contexts.