Problem 14 - Algebra 1

Question

Use the distributive property and mental math to simplify the expression. $$ 6 x^{2}-4 x^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^{2}\).

Step 1: Identify Like-Terms

The goal is to simplify the expression. Look for like-terms to combine. Here, \(6x^{2}\) and \(-4x^{2}\) are like-terms because they both contain the variable \(x^{2}\).

Step 2: Combine Like-Terms

Now, apply the distributive property to combine these like terms. This is done by subtracting the coefficients. The coefficient of \(6x^{2}\) is 6 and of \(-4x^{2}\) is -4. Subtract -4 from 6 to get 2.

Step 3: Final Simplified Expression

The simplified expression becomes \(2x^{2}\) as a result. Remember, when combining like-terms, the variable part (\(x^{2}\)) stays the same. You only add or subtract the coefficients.

Key Concepts

Understanding Like-Terms

In algebra, identifying like-terms is a fundamental concept when simplifying expressions. Like-terms are terms that share the same variable and exponent, which means they can be combined through addition or subtraction. For example, in the expression given, \(6x^{2}\) and \(-4x^{2}\) are like-terms. They both have the same variable, \(x\), raised to the same power, which is \(^2\). This allows us to simplify the expression by combining them.To combine like-terms, focus only on the numerical coefficients and perform the relevant arithmetic operation:

• Look for terms that have identical variable parts.
• Add or subtract their coefficients to combine them.
• Keep the variable part unchanged.

When you apply these steps, you simplify the expression by reducing it to fewer terms while maintaining its original value.

The Power of Mental Math

Mental math refers to the ability to carry out arithmetic calculations in your head, without the need for paper, pencil, or a calculator. Using mental math in algebra can make solving equations and simplifying expressions faster and more intuitive. In our example, after identifying like-terms, you need to subtract the coefficients:

• The coefficient of \(6x^{2}\) is 6.
• The coefficient of \(-4x^{2}\) is -4.
• Subtract -4 from 6 mentally, which gives you 2.

This mental subtraction is simple yet crucial, making the process of combining like-terms and simplifying the expression more efficient.Practicing mental math regularly enhances your numeracy skills, making it easier to handle more complex algebra problems with confidence.

Mastering Simplifying Expressions

Simplifying expressions is all about reducing the expression to its most concise form while preserving its equivalence. This process typically involves combining like-terms, as shown in our algebraic expression example, \(6x^{2} - 4x^{2}\).Here's a simple plan for simplifying:

• Identify like-terms: Examine each term in the expression to find variable and exponent matches.
• Combine coefficients: Perform the necessary arithmetic (addition or subtraction) to combine the coefficients of like-terms.
• Write the simplified expression: Use the combined coefficient with the retained variable part.

In our example, the simplified expression \(2x^{2}\) still retains the same variable part but is reduced to one single term by combining like-terms. Simplifying expressions makes them easier to work with in equations and provides a clearer understanding of the mathematical relationships.