Problem 96 - Intermediate Algebra

Question

Simplify each expression. \(3\left(2 a^{2}-3 a+2\right)-4\left(2 a^{2}+4 a-7\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(-2a^2 - 25a + 34\).

Step 1: Expand Each Bracket

Start by distributing each number outside the brackets to each term inside the brackets. For the first expression, distribute 3 to each term of \(2a^2 - 3a + 2\), resulting in \(3 \times 2a^2 = 6a^2\), \(3 \times (-3a) = -9a\), and \(3 \times 2 = 6\). Similarly, distribute -4 to each term of \(2a^2 + 4a - 7\), resulting in \(-4 \times 2a^2 = -8a^2\), \(-4 \times 4a = -16a\), and \(-4 \times (-7) = 28\).

Step 2: Write the Expanded Expression

Combine the expanded terms from Step 1 into a single expression: \(6a^2 - 9a + 6 - 8a^2 - 16a + 28\).

Step 3: Combine Like Terms

Combine the like terms by grouping them together. For the \(a^2\) terms, combine \(6a^2\) and \(-8a^2\) to get \(-2a^2\). For the \(a\) terms, combine \(-9a\) and \(-16a\) to get \(-25a\). For the constant terms, combine \(6\) and \(28\) to get \(34\).

Step 4: Write the Simplified Expression

The simplified expression after combining like terms from Step 3 is \(-2a^2 - 25a + 34\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics, particularly in algebra. It allows us to simplify expressions by breaking down complex expressions into simpler ones. The property states that for any numbers or expressions, if you have to multiply a set of terms inside a bracket by a number, you can do this by multiplying each term inside the bracket by the number outside.

• For example, in the expression \(3(2a^2 - 3a + 2)\), we distribute the 3 to each term inside the brackets: \(3 \times 2a^2\), \(3 \times (-3a)\), and \(3 \times 2\).

This makes complex multiplication tasks more manageable by turning them into a series of simple multiplication tasks. If we do the same for another part of the original expression \(-4(2a^2 + 4a - 7)\), we multiply each term inside the brackets by -4, giving us \(-4 \times 2a^2\), \(-4 \times 4a\), and \(-4 \times (-7)\).
Using the distributive property simplifies the process of manipulating algebraic expressions and is a key step in solving equations as it facilitates the subsequent processes of combining like terms.

Combining Like Terms

Once you've used the distributive property to expand an expression, the next step is to combine like terms. Combining like terms is the process of merging terms with the same variables raised to the same powers. This step simplifies the expression even further.

• In our original problem, after using the distributive property, we end up with the expanded expression: \(6a^2 - 9a + 6 - 8a^2 - 16a + 28\).
• The like terms here are \(a^2\) terms, \(a\) terms, and constant numbers.

You combine the \(a^2\) terms: \(6a^2\) and \(-8a^2\) to get \(-2a^2\). For the \(a\) terms: \(-9a\) and \(-16a\), which results in \(-25a\). Lastly, the constant terms: 6 and 28 combine to 34.
Combining like terms is a crucial operation because it reduces the expression to its simplest form and prepares it for further analysis or for solving an equation. By logically organizing these terms, you can handle algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra and appear in various forms, such as linear, quadratic, or polynomial expressions. Knowing how to manipulate these expressions is key to mastering algebra.

• An algebraic expression like \(3(2a^2 - 3a + 2) - 4(2a^2 + 4a - 7)\) consists of various terms that can include variables raised to different powers and constants.
• An expression must often be simplified or restructured to solve problems or to see relationships more clearly.

The essence of working with algebraic expressions lies in understanding how to apply rules like the distributive property and how to combine like terms. These skills allow you to convert a complex expression into a simpler one like \(-2a^2 - 25a + 34\).
Mastering algebraic expressions also sets a solid foundation for studying equations, functions, and more advanced math topics. With practice, these expressions become tools for solving real-life problems and exploring mathematical concepts.