Problem 45 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(-a^{2}+1\right)-\left(a^{2}-3\right)+\left(5 a^{2}-6 a+7\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(3a^2 - 6a + 11\).

Step 1: Expand the Expression

Write out the expression with proper brackets to ensure correct order of operations: \[(-a^2 + 1) - (a^2 - 3) + (5a^2 - 6a + 7)\] Expand each set of parentheses:\[-a^2 + 1 - a^2 + 3 + 5a^2 - 6a + 7\]

Step 2: Combine Like Terms

Group and combine like terms. We have three kinds of terms: constant, terms with \(a\), and terms with \(a^2\).- Combine \(a^2\) terms: \(-a^2 - a^2 + 5a^2 = 3a^2\)- Combine constant terms: \(1 + 3 + 7 = 11\)- Combine \(a\) terms (note \(-6a\) is the only term): \(-6a\)The simplified expression is:\[3a^2 - 6a + 11\]

Key Concepts

Order of Operations

When solving mathematical expressions, following the order of operations is crucial. It ensures that everyone calculates expressions consistently and correctly. The standard rule for the order of operations is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right).

In the context of algebraic expressions, like the one given in the original exercise, parentheses play a vital role. They control which operations should be performed first. By expanding the parentheses in the expression \((-a^2 + 1) - (a^2 - 3) + (5a^2 - 6a + 7)\), we prioritize what's inside the brackets before performing additional operations like combining terms. This step helps avoid mistakes and assures the final solution is accurate.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms that contain the same variables raised to the same power. For example, in the given exercise:

• \(a^2\) terms: \(-a^2\), \(-a^2\), and \(5a^2\)
• \(a\) terms: There is only one, \(-6a\)
• Constant terms: \(1\), \(3\), and \(7\)

By adding or subtracting these like terms together, we can simplify the expression. So, for the \(a^2\) terms in our expression \(-a^2 - a^2 + 5a^2\), we can combine them into \(3a^2\). Similarly, the constant terms \(1 + 3 + 7\) combine to become \(11\). Since \(-6a\) stands alone as the only \(a\) term, it remains unchanged in the final expression. The simplified version is \(3a^2 - 6a + 11\). This process not only reduces complexity but also makes the expression easier to analyze and utilize.

Algebraic Expressions

An algebraic expression includes numbers, variables, and operators like addition, subtraction, multiplication, and division. In the exercise, each part of the expression \((-a^2 + 1) - (a^2 - 3) + (5a^2 - 6a + 7)\) is an algebraic expression.

To work effectively with these expressions, understanding components such as coefficients, constants, and terms is important:

• **Coefficients** are numbers that multiply the variables. For example, in \(5a^2\), \(5\) is the coefficient.
• **Constants** are plain numbers not attached to any variables, like \(1\), \(3\), and \(7\).
• **Terms** are individual parts of an expression separated by a plus or minus sign. In the expanded form \(-a^2 + 1 - a^2 + 3 + 5a^2 - 6a + 7\), there are multiple terms like \(-a^2\) and \(7\).

Understanding these components lays the foundation for manipulating expressions, solving equations, and eventually working with more advanced mathematics. Mastery of algebraic expressions simplifies learning and boosts confidence in handling mathematical problems.