Problem 63 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((x+3 y)(x-3 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(x^2 - 9y^2\).

Step 1: Identify the Special Product Formula

The expression \((x+3y)(x-3y)\) can be identified as a difference of squares. The difference of squares formula is given by \((a+b)(a-b) = a^2 - b^2\). In our expression, \(a = x\) and \(b = 3y\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula \((a^2 - b^2)\), substitute \(a = x\) and \(b = 3y\) to get the simplified form:\[ (x+3y)(x-3y) = x^2 - (3y)^2 \]

Step 3: Simplify the Expression

Simplify the expression by evaluating \((3y)^2\):\[(3y)^2 = 9y^2\]Now substitute back into the expression:\[x^2 - 9y^2\]

Key Concepts

Special Product Formula

The Special Product Formula is a handy math tool that helps us easily compute the product of particular algebraic expressions. A very common type is the "Difference of Squares". This formula is expressed as \((a+b)(a-b) = a^2 - b^2\). It means that when you have two binomials with identical terms but opposite signs, their product will always be a simple calculation. This special product identity saves us from performing lengthy distribution.

• Recognizing when to use this formula is key. If you see expressions like \((x+3y)(x-3y)\), you can jump straight to writing it as a difference of squares.
• By applying the formula, \((a+b)(a-b)\) becomes \(a^2 - b^2\).
• This formula reduces computation time and minimizes possible errors in calculation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). In our exercise, we have the expressions \((x+3y)\) and \((x-3y)\). Each part can be seen as both a potential target for simplification and an opportunity to apply algebraic rules.
When dealing with algebraic expressions, it's important to identify:

• Variables and constants: These are your basic building blocks. Here, variables like \(x\) and \(y\) act as placeholders for numbers.
• Operations: These define how your variables interact. In this case, addition and subtraction set the stage for the difference of squares.
• Grouping: Parentheses tell you which calculations to focus on first.

Understanding these expressions allows you to apply formulas like the difference of squares effectively.

Simplifying Expressions

Simplifying expressions means making them as compact as possible while keeping the same value. For the problem \((x+3y)(x-3y)\), once we apply the difference of squares formula, we need to simplify further to make it neater.
In this expression, you simplify by:

• First applying the result of the formula: \((x+3y)(x-3y) = x^2 - (3y)^2\).
• Breaking down components: Calculate \((3y)^2\). Here, \((3y)^2\) becomes \(9y^2\), simplifying each term within.
• Arriving at final compact form: Substitute \(9y^2\) back into the formula to get \(x^2 - 9y^2\).

This streamlined process not only leads to a cleaner answer but also reinforces understanding of how each step works alongside algebraic principles. Keeping track of each element as it's simplified ensures accuracy in solving algebra problems.