Problem 69 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(9 a^{2}+3 a\right)-\left(2 a-4 a^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 13a^2 + a \).

Step 1: Distribute Negative Sign

The given expression is \( (9a^2 + 3a) - (2a - 4a^2) \). Start by distributing the negative sign across the second set of parentheses: \( -1 \times (2a - 4a^2) = -2a + 4a^2 \). Now, rewrite the expression without the parentheses: \( 9a^2 + 3a - 2a + 4a^2 \).

Step 2: Combine Like Terms

Combine the like terms from the expression \( 9a^2 + 3a - 2a + 4a^2 \). The like terms \( 9a^2 \) and \( 4a^2 \) can be combined to get \( (9a^2 + 4a^2) = 13a^2 \). Similarly, combine the terms \( 3a \) and \( - 2a \) to get \( (3a - 2a) = 1a \).

Step 3: Write the Final Simplified Expression

Combine all the simplified terms from the earlier steps: \( 13a^2 + 1a \). Simplify the expression further by omitting the coefficient of 1: \( 13a^2 + a \). This is the simplified result of the original operation.

Key Concepts

Distributive Property

The distributive property is an essential concept in mathematics that helps in solving expressions involving parentheses. In the given exercise, we initially encounter an expression

• \( (9a^2 + 3a) - (2a - 4a^2) \)

To eliminate the parentheses, we utilize the distributive property by distributing the negative sign across the terms contained inside the second set of parentheses. Distributing a negative sign is equivalent to multiplying each term inside the parentheses by -1:

• \( -1 \times (2a - 4a^2) = -2a + 4a^2 \)

After distribution, the expression is rewritten without parentheses:

• \( 9a^2 + 3a - 2a + 4a^2 \)

This clarification removes the grouping brought by parentheses and prepares the equation for the next steps like combining like terms.

Combining Like Terms

When working with polynomials, combining like terms is a crucial step to simplify expressions effectively. Like terms are terms in an expression that have the same variables raised to the same powers. In other words, you can only combine terms that "look alike" with respect to their variables and exponents.
In the expression \( 9a^2 + 3a - 2a + 4a^2 \), observe that:

• \( 9a^2 \) and \( 4a^2 \) are like terms because they both contain the variable \( a \) raised to the second power.
• Similarly, \( 3a \) and \(-2a\) are like terms since they have the variable \( a \) raised to the first power.

Now combine these terms:

• Add \( 9a^2 \) and \( 4a^2 \) to get \( 13a^2 \).
• Add \( 3a \) and \(-2a\) to get \( 1a \), often written simply as \( a \).

By successfully combining like terms, we are closer to fully simplifying the expression.

Simplifying Expressions

Simplifying mathematical expressions involves conducting operations that reduce the expression into its simplest form. After utilizing the distributive property and combining like terms, we reach the simplified polynomial form of an expression.

• Our initial steps provided the expression \( 13a^2 + a \), which cannot be simplified further because there are no more like terms.

The key to simplifying expressions is conducting all possible reductions without altering the original value of the expression. This often includes:

• Eliminating parentheses through distribution if needed.
• Combining like terms as demonstrated.
• Reducing any coefficients and eliminating unnecessary zeros or ones, such as rewriting \( 1a \) as simply \( a \).

Always remember the goal of simplification is to achieve a cleaner, more understandable expression that retains its original equivalency. Therefore, the solution \( 13a^2 + a \) represents the most simplified and readily usable form.