Problem 59 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((x+5)(x-5)\)

Step-by-Step Solution

Answer

The simplified expression is \(x^2 - 25\).

Step 1: Identify the Special Product Formula

The expression \((x+5)(x-5)\) is an example of the difference of squares. The formula for the difference of squares is \((a+b)(a-b) = a^2 - b^2\), where \(a\) and \(b\) are any two terms. In this case, \(a\) is \(x\) and \(b\) is \(5\).

Step 2: Apply the Difference of Squares Formula

Now that we've identified \(a = x\) and \(b = 5\), substitute these into the formula: \((a+b)(a-b) = a^2 - b^2\). Therefore, \((x+5)(x-5) = x^2 - 5^2\).

Step 3: Simplify the Expression

Calculate the squares: \(x^2\) stays as \(x^2\) and \(5^2 = 25\). Thus, the expression becomes \(x^2 - 25\).

Key Concepts

Special Product Formula

The special product formula is a handy tool in algebra that helps simplify certain types of expressions. One such formula is the difference of squares, which applies when you have a binomial of the form

• (a + b)(a - b).

The core idea is that multiplying these binomials results in a very specific pattern:

• \(a^2 - b^2\).

This formula saves you from manually expanding and simplifies the multiplication process. By recognizing patterns, such as

• \((x+5)(x-5)\),

you can directly apply the formula to quickly reduce the expression to a much simpler form, which is

• \(x^2 - 25\).

Understanding these shortcuts not only makes solving such problems faster but also reinforces your grasp of the underlying algebraic principles.

Algebraic Expressions

Algebraic expressions are a crucial aspect of algebra, where you work with symbols and numbers. These expressions can include variables, constants, and mathematical operations like addition, subtraction, multiplication, and division. In the case of the exercise

• \((x+5)(x-5)\),

we are dealing with expressions involving the variable

• x.

Each expression represents a number that can change depending on the value assigned to the variable. Algebra helps in finding relationships and solving problems by manipulating these expressions. Recognizing forms like

• the difference of squares,

allows for easier and quicker computation by using known formulas. This appreciation for expression forms boosts your mathematical fluency and problem-solving capabilities.

Simplification Steps

Simplification involves reducing expressions to their simplest form. This makes equations easier to work with. As shown in our exercise

• \((x+5)(x-5)\),

applying the special product formula transforms it directly into

• \(x^2 - 5^2\).

Next, simplification entails computing any basic operations such as multiplication or squaring.
For example, squaring a number means multiplying it by itself, so

• \(5^2 = 25\).

This results in the final simplified expression:

• \(x^2 - 25\).

Effective simplification requires recognizing patterns and performing operations neatly, ensuring all terms are calculated correctly.
Through consistent practice, these simplification skills become an essential part of your algebra toolkit.