Problem 95 - Intermediate Algebra

Question

Simplify each expression. $$6.4 a^{2}+11.8 a-9.2 a+5.7$$

Step-by-Step Solution

Answer

The expression simplifies to \(6.4a^{2} + 2.6a + 5.7\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. We have terms involving the variable \(a\) and constant terms. Here, the terms \(11.8a\) and \(-9.2a\) are like terms.

Step 2: Combine Like Terms

Combine the like terms. Add \(11.8a\) and \(-9.2a\) together: \(11.8a - 9.2a = 2.6a\).

Step 3: Write the Simplified Expression

After combining like terms, the expression simplifies to: \(6.4a^{2} + 2.6a + 5.7\). This is the most simplified form as there are no further like terms to combine.

Key Concepts

Understanding Like Terms

In any polynomial expression, like terms are terms that have the same variables raised to the same power. The coefficients (numbers in front of the variables) can be different. For example, in the expression given, both \(11.8a\) and \(-9.2a\) are like terms because they contain the same variable, \(a\), raised to the same power (which is 1 in this case). Meanwhile, \(6.4a^2\) and \(5.7\) are not like terms with either \(11.8a\) or \(-9.2a\), as their variables and powers differ.

• Terms like \(a^2\), \(a\), and constant terms (like \(5.7\)) cannot be combined with each other.
• It's important to identify like terms, as only these can be combined to simplify expressions.

Combining Like Terms

Combining like terms is the process of adding or subtracting them, based on their coefficients, to simplify the expression. In the original exercise, we have identified \(11.8a\) and \(-9.2a\) as like terms. By performing the subtraction, \(11.8a - 9.2a\), we combine these into a single term: \(2.6a\).

• Align like terms by their variable and power before combining.
• Make sure to carry any negative signs through the arithmetic to avoid errors.

This process does not affect the terms that are not alike, such as \(6.4a^2\) or \(5.7\), and they remain unchanged in this step.

Simplifying Expressions

When simplifying expressions, the goal is to condense the expression as much as possible by performing arithmetic on the coefficients of like terms. In our exercise, we identified and combined like terms to simplify the original expression to \(6.4a^2 + 2.6a + 5.7\). This step-by-step simplification ensures that the expression is in its simplest form, making it easier to understand or to substitute into further equations.

• Always ensure that no like terms are left uncombined.
• You can further verify your work by checking that no further identical terms are left, like terms have been combined correctly, and all arithmetic operations were performed accurately.

Simplification is an essential skill in algebra, helping to solve equations more efficiently and clearly.