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Chapter 1: Problem 89

Translate to an algebraic expression. Half of the difference of the squares of two numbers

Short Answer

Expert verified
\( \frac{1}{2} (x^2 - y^2) \)

Step by step solution

01

Identify the Variables

Let the two numbers be represented by the variables x and y.
02

Write the Squares of the Numbers

The squares of the numbers are \[ x^2 \] and \[ y^2 \].
03

Calculate the Difference of the Squares

The difference of the squares is given by \[ x^2 - y^2 \].
04

Calculate Half of the Difference

Half of the difference of the squares is given by \[ \frac{1}{2} (x^2 - y^2) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variables
In algebra, a variable is a symbol used to represent an unknown number. Variables are typically denoted by letters such as x, y, or z. In our exercise, we use variables to stand in for the two unknown numbers. By doing this, we can translate real-world problems into algebraic expressions that are easier to manipulate and solve.
When you see a variable, think of it as a placeholder that can take any value. This flexibility is what makes algebra so powerful.
Squares of Numbers
The square of a number is the result of multiplying the number by itself. It's represented as \[x^2\] for a number x. Squaring is a fundamental operation in algebra because it helps us describe areas, compute distances, and understand complex relationships between quantities.

For example:
  • \[(2)^2 = 4\]
  • \[(3)^2 = 9\]
  • \[(x)^2 = x \times x = x^2\]

In our exercise, we square our variables to start forming the needed expression.
Difference of Squares
The difference of squares is a mathematical concept that involves subtracting one square from another. It's expressed as \[x^2 - y^2\]. This form appears often in algebra and can be factored into \[(x+y)(x-y)\].
This is useful because factoring can simplify expressions and solve equations more easily. Let's look at an example:
  • \[9 - 4 = (3^2 - 2^2) = (3 + 2)(3 - 2) = 5 \times 1 = 5\]

In our exercise, we use this concept to find the difference between the squares of two variables.
Fractions
A fraction represents a part of a whole and is written as one number over another, separated by a line. For example, \[\frac{1}{2}\] represents one half.
In algebra, fractions can be used to scale values - for instance, finding half of something might use \[\frac{1}{2} \].

To find half of the difference of the squares, we've derived the expression \[\frac{1}{2}(x^2 - y^2)\]. This lets us neatly express 'half of' a more complex operation, showing how fractions simplify algebraic manipulations.

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Most popular questions from this chapter

Translate to an algebraic expression. \(\\{x | x=5 n, n \text { is a natural number }\\}\)

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