Problem 67 - Elementary Algebra Within Reach

Question

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ \text { The square of } x \text { decreased by the product of } x \text { and } 2 x $$

Step-by-Step Solution

Answer

The algebraic expression translates to \(-x^2\).

Step 1: Translation

Translate the verbal description into a mathematical expression. Here 'The square of x' can be written as \(x^2\). 'Product of x and 2x' is \(2x^2\). According to the phrase, the latter is to be subtracted from the former. So, put together it forms \(x^2 - 2x^2\).

Step 2: Simplification

Simplify the equation using the basic principle of algebra. Subtracting 2x^2 from x^2 equals -x^2.

Key Concepts

Simplification

Simplification in algebra involves taking an expression and reducing it to its simplest form. By simplifying algebraic expressions, we aim to make them easier to understand and work with in equations. In our example, this involves a common action in algebra: combining like terms.

Let's break it down further:

• Identify like terms: These are terms in the expression that contain the same variables raised to the same power. In the equation from the exercise, both terms are multiples of \(x^2\).
• Perform the operation: Once like terms are identified, you combine them. In this case, subtract \(2x^2\) from \(x^2\), which results in \(-x^2\).

Through this process, we simplify the expression \(x^2 - 2x^2\) to \(-x^2\). Simplifying expressions can often make solving equations much clearer and give insight into the behavior of the equation.

Translation into Algebraic Form

Translation of verbal phrases into algebraic expressions is a crucial step in understanding and solving math problems. This transformation requires careful interpretation of the words used.

For instance, the phrase "The square of \(x\) decreased by the product of \(x\) and \(2x\)" can be broken down as follows:

• "The square of \(x\)": This part instructs us to write \(x^2\).
• "the product of \(x\) and \(2x\)": Here, "product" signals multiplication, resulting in \(2x^2\).
• "decreased by": This phrase denotes subtraction.

By translating each component of the verbal description, we get the algebraic expression \(x^2 - 2x^2\). This process allows complex verbal problems to be converted into mathematical forms that are easier to manipulate.

Verbal Phrases in Math

Verbal phrases in math can often seem tricky, especially because the language used can vary. However, breaking them down into simple actions can make the task much easier. Understanding common phrases and terms is key.

Consider some common verbal phrases and their algebraic meanings:

• "The sum of": Indicates addition.
• "The difference between": Refers to subtraction.
• "The product of": Signifies multiplication.
• "The quotient of": Denotes division.
• "Increased by" or "more than": Also pointers to addition.

In the context of the example problem, "decreased by" indicated the need for subtraction. With consistent practice, recognizing and translating these phrases becomes second nature, allowing equations to form from verbal descriptions seamlessly, paving the way for easier problem solving.