Problem 68 - Elementary Algebra Within Reach

Question

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ \text { The square of } x \text { added to the product of } x \text { and } x+1 $$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(2x^2 + x\).

Step 1: Convert the Phrase to Algebraic Expression

The verbal phrase can be translated into an algebraic expression as follows: 'The square of \(x\)' is represented by \(x^2\). 'Added to' indicates a plus operation. 'The product of \(x\)' and '\(x+1\)' refers to the multiplication of \(x\) with \(x+1\). So, the full expression becomes \(x^2 + x*(x+1)\).

Step 2: Simplify the Algebraic Expression

To simplify the expression, distribute \(x\) to \(x+1\) which gives \(x^2 + x*x + x*.1\). This equals \(x^2 + x^2 + x = 2x^2 + x\).

Step 3: Final Simplified Expression

The simplified expression is \(2x^2 + x\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes complex problems easier to solve. To simplify an expression means to combine like terms and carry out the operations required to make the expression as concise as possible.

For example, in the expression \( x^2 + x(x+1) \), each term represents a group of similar variables or constants. To simplify it, you perform the arithmetic operations:

• Distribute: Apply the distributive property by multiplying \(x\) with each term inside the parenthesis, \(x+1\), which results in \(x \cdot x + x \cdot 1 = x^2 + x\).
• Combine like terms: In our expression, both \(x^2\) terms can be combined since they involve the same degree of \(x\). Therefore, \(x^2 + x^2 + x\) simplifies to \(2x^2 + x\).

The outcome is a significantly more straightforward expression that is easier to work with in further algebraic operations or problem-solving scenarios.

Translating Verbal Phrases

Translating verbal phrases to algebraic expressions involves converting words into mathematical symbols and operations. This skill is crucial because it allows you to turn real-world situations or word problems into solvable math equations.

Consider a phrase like 'the square of \(x\) added to the product of \(x\) and \(x+1\)'. Here's how to translate this into an algebraic expression:

• Identify 'the square of \(x\)' as \(x^2\) since squaring involves multiplying \(x\) by itself.
• Recognize 'added to' as a signal to use the plus sign \(+\) in the expression.
• Understand 'the product of \(x\) and \(x+1\)' as \(x \cdot (x+1)\), which means you multiply \(x\) by the sum \((x+1)\).

When you piece these parts together, the phrase converts to the expression \(x^2 + x(x+1)\). By learning to identify these key terms and operations, you can translate any verbal statement into algebraic terms effectively.

Algebraic Operations

Algebraic operations are the basic mathematical manipulations applied to algebraic expressions. These operations include addition, subtraction, multiplication, division, and sometimes exponentiation. Each has specific rules guiding how they should be performed.

In the context of our expression, \(x^2 + x(x+1)\), several operations are involved:

• The multiplication \(x(x+1)\) utilizes the distributive property, which states you multiply the term outside the parenthesis by each term inside, leading to \(x^2 + x\).
• Addition is used to combine the terms after simplification, resulting in the final expression \(2x^2 + x\).

By mastering these operations, you develop the flexibility to manipulate and simplify expressions effectively, paving the way to solve more complex algebraic problems.