Problem 61 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. the square of 14 less than a number b. 14 less than the square of a number

Step-by-Step Solution

Answer

(a) \((x - 14)^2\), (b) \(x^2 - 14\)

Step 1: Understand the Phrase

For part (a), the phrase 'the square of 14 less than a number' means you need to subtract 14 from a number first, and then square the result. For part (b), the phrase '14 less than the square of a number' means you first square the number and then subtract 14.

Step 2: Define a Variable

Let's define a variable to represent 'a number'. We'll use \( x \) to denote this unknown number.

Step 3: Part (a) - Set Up the Expression for "the square of 14 less than a number"

From Step 1(a), subtract 14 from \( x \) first: \( x - 14 \). Then, square the result to get: \[ (x - 14)^2 \].

Step 4: Part (b) - Set Up the Expression for "14 less than the square of a number"

From Step 1(b), first square \( x \): \( x^2 \). Then, subtract 14 from that result to get: \[ x^2 - 14 \].

Key Concepts

Variable Substitution

In algebra, we often encounter scenarios where we need to represent unknown quantities. In such cases, we use variables. A variable is like a placeholder, typically represented by a letter like \( x \). It stands in for a number we don't know yet.
In our given exercise, we have to translate phrases into algebraic expressions. Both parts 'a' and 'b' refer to "a number" which isn’t specified. Thus, we can use the variable \( x \) for this purpose. This is called "variable substitution" where we replace a described or unknown quantity with a variable.
By assigning variables, we simplify the process of working with and manipulating these expressions. It’s a crucial step in setting up our algebraic equations and finding solutions. Remember, any letter or symbol can serve as a variable, but \( x \) is just the most common choice.

Expression Translation

Translating a phrase into an algebraic expression involves understanding the relationships and operations described in words. We break down the phrase to identify what mathematical operation needs to be done first.
For instance, in part (a), the phrase "the square of 14 less than a number" means:

• First, we identify "14 less than a number." This requires us to subtract 14 from our variable \( x \): \( x - 14 \).
• Then, we take the result and square it. Thus, the expression becomes \( (x - 14)^2 \).

In part (b), the phrase "14 less than the square of a number" identifies a different sequence:

• First, square the variable \( x \): \( x^2 \).
• Then, subtract 14: \( x^2 - 14 \).

Methodically working through each part of the phrase helps us correctly translate words into mathematical language.

Algebraic Operations

Once we have our expression set up according to the phrase, we need to perform algebraic operations. Operations like addition, subtraction, multiplication, and exponentiation are the building blocks of algebra.
To compute or simplify any algebraic expression, understanding these operations is key. In the given exercise:

• For \( (x - 14)^2 \): This involves first executing the operation of subtraction, leaving us with \( x - 14 \). Then we perform exponentiation by squaring the result.
• For \( x^2 - 14 \): We begin with the operation of squaring \( x \) to obtain \( x^2 \). Then, we perform subtraction by taking away 14 from the squared variable.

Mastery of basic operations allows us to manipulate expressions, solve equations, and understand algebraic relationships. Each operation follows specific rules and order, ensuring that we obtain accurate results from our calculations.