Problem 68 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ x(x+2)+2\left(x^{2}+3 x-4\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(\displaystyle x\left(x +2\right) +2\left(x^{2} +3x-4\right)\). Answer: \(\displaystyle 3x^{2} + 8x - 8\)

Step 1: Distribute the first term

We will first distribute the x term across the first bracket: $$ x(x+2) = x^2 + 2x $$ Now our expression becomes: $$ (x^2 + 2x) + 2\left(x^{2}+3 x-4\right) $$

Step 2: Distribute the second term

Now, we will distribute the 2 across the second bracket: $$ 2\left(x^{2}+3 x-4\right) = 2x^2+6x-8 $$ Our expression is now in the form: $$ (x^2 + 2x) + (2x^2+6x-8) $$

Step 3: Combine like terms

Now, we will combine our like terms: $$ x^2+2x+2x^2+6x-8 $$ Combine the \(x^2\) terms: $$ x^2+2x^2=3x^2 $$ Combine the x terms: $$ 2x+6x=8x $$ Our simplified expression now becomes: $$ 3x^2 + 8x - 8 $$ Thus, the simplified algebraic expression is \(\displaystyle 3x^{2} +8x-8\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a key skill in algebra that helps make expressions easier to understand and solve. To simplify an expression means to reduce it to its simplest form without changing its value. This often involves a series of operations such as distributing, combining like terms, and eliminating unnecessary parts of the expression.

When simplifying expressions, always follow the proper order of operations. Look for opportunities to apply properties like the distributive property or combine any like terms. This is done to consolidate and reduce the expression to a form that is easier to work with. Simplification is not just about making expressions shorter but also about making them clearer and less prone to errors in further calculations.

Distributive Property

The distributive property is a fundamental algebraic concept used to break down expressions for easier handling. It allows you to multiply a single term by each term inside a parenthesis. The property is expressed mathematically as: \[a(b + c) = ab + ac\] This property ensures that each term inside the parenthesis is multiplied by the term outside. In our exercise, we use the distributive property twice.

First, multiply \(x\) with \(x + 2\) to get \(x^2 + 2x\). Then, apply it again with the second set of parentheses by multiplying \(2\) by each term within \(x^2 + 3x - 4\), resulting in \(2x^2 + 6x - 8\). This step ensures that every part of the expression is expanded, setting the stage for the next step of simplification.

Combining Like Terms

Combining like terms is an essential process in simplifying expressions. It involves grouping terms in an expression that have the same variable and exponent, known as 'like terms'. For example, \(3x^2\) and \(4x^2\) are like terms because they both contain the variable \(x^2\).

In the exercise, after distributing, you get two expressions: \(x^2 + 2x\) and \(2x^2 + 6x - 8\). To combine like terms, you first add together the \(x^2\) terms from both expressions: \(x^2 + 2x^2\) results in \(3x^2\). Then, add the \(x\) terms: \(2x + 6x = 8x\). The constant \(-8\) remains unchanged since it has no like terms in this case.

This critical step helps to condense the expression into its simplest form, here being \(3x^2 + 8x - 8\), making further operations easier and clearer.