Problem 62 - Elementary Algebra

Question

Simplify. $$ a 4-a+a 2-9 a+18 a 2-13 a+36 $$

Step-by-Step Solution

Answer

19a^2 - 23a + 40

Step 1: Identify Like Terms

The expression is composed of terms like \(a\), \(a^2\), and constants. First, list all similar terms: \(a\), \(a^2\), and constants are 4 and 36.

Step 2: Combine Like Terms for \(a^2\)

There are two \(a^2\) terms: \(a^2\) and \(18a^2\). Combine them together: \(a^2 + 18a^2 = 19a^2\).

Step 3: Combine Like Terms for \(a\)

There are four \(a\) terms: \(-a, -9a, -13a\). Combine them: \(-a - 9a - 13a = -23a\).

Step 4: Combine Constant Terms

There are two constant terms: 4 and 36. Combine them: \(4 + 36 = 40\).

Step 5: Write Simplified Expression

Combine the results from the previous steps to form the simplified expression: \(19a^2 - 23a + 40\).

Key Concepts

Like Terms

In algebra, when dealing with expressions, it's crucial to understand the concept of **like terms**. Like terms are those that contain the same variable raised to the same power. This means their variables and exponents must match exactly. For example, in the expression given:

• Terms combined with a variable "a" are like terms, such as \(-a\), \(-9a\), and \(-13a\).
• Terms like \(a^2\) and \(18a^2\) are also like terms because they both involve the variable "a" squared.

Identifying like terms is a fundamental step in simplifying algebraic expressions. It allows us to combine them easily by adding or subtracting their coefficients. This process makes managing algebraic manipulations simpler and clearer.

Algebraic Expressions

An **algebraic expression** is a mathematical phrase that contains variables, numbers, and operations. It does not include an equal sign and is not a complete equation or inequality. Looking at the problem, it involves several components:

• **Variables**: These are symbols that represent numbers, typically letters such as "a" in the expression.
• **Coefficients**: Numbers that multiply the variables, for instance, \(18\) is the coefficient of \(a^2\).
• **Constants**: These are numbers on their own, like \(4\) and \(36\).

Understanding each part of an algebraic expression helps in reading and solving algebra problems. Our goal is usually to simplify these expressions by reducing them to their most basic form by using operations like addition and subtraction.

Combining Terms

**Combining terms** is the process of adding or subtracting like terms in an algebraic expression. This activity simplifies the expression and makes it easier to interpret or solve. Here's how we simplify the given expression:

• Start with the \(a^2\) terms: \(a^2\) and \(18a^2\). We combine them to get \(19a^2\) by adding their coefficients (1 + 18 = 19).
• Next, address the terms with the variable \(a\): \(-a\), \(-9a\), and \(-13a\). Combining them gives us \(-23a\) by adding the coefficients (-1 - 9 - 13 = -23).
• Finally, add the constant terms: \(4\) and \(36\), resulting in \(40\) because \(4 + 36 = 40\).

Each of these steps reduces complexity, allowing us to translate a tangled expression into its simplest form: \(19a^2 - 23a + 40\). Simplifying ensures that expressions are presented in their clearest, most manageable state.