Problem 124 - Elementary Algebra

Question

For the following problems, perform the indicated operations and combine like terms. $$ 6 x^{2}-12 x+\left(4 x^{2}-3 x-1\right)+4 x^{2}-10 x-4 $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $6 x^{2}-12 x+\left(4 x^{2}-3 x-1\right)+4 x^{2}-10 x-4$ Answer: $14x^2 - 25x - 5$

Step 1: Identify like terms in the given expression

In the given expression: $$ 6 x^{2}-12 x+\left(4 x^{2}-3 x-1\right)+4 x^{2}-10 x-4 $$ we can see the like terms are: - \(x^2\) terms: \(6x^2\), \(4x^2\), and \(4x^2\) - \(x\) terms: \(-12x\), \(-3x\), and \(-10x\) - constant terms: \(-1\), and \(-4\)

Step 2: Combine like terms

Add or subtract the like terms: \(x^2\) terms: $$ 6x^2 + 4x^2 + 4x^2 = 14x^2 $$ \(x\) terms: $$ -12x - 3x - 10x = -25x $$ constant terms: $$ -1 - 4 = -5 $$

Step 3: Write the simplified expression

Now, combine the simplified terms to get the final simplified expression: $$ 14x^2 - 25x - 5 $$ So the simplified expression is: $$ 14x^2 - 25x - 5. $$

Key Concepts

Algebraic Expressions

Algebraic expressions represent a combination of numbers, variables (like x or y), and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions can take many forms, from simple terms like 5x to more complex equations.

For instance, the expression in the exercise,
\(6 x^{2}-12 x+(4 x^{2}-3 x-1)+4 x^{2}-10 x-4\),
involves several terms with different powers of x, constants, and arithmetic operations. Understanding the structure of this algebraic expression is crucial before one can begin simplifying it.

Each term in an algebraic expression can act independently or interact with other terms. The complexity of algebraic expressions varies, and sometimes it may require several steps to simplify them, especially when parentheses are involved. This is why it's important to identify and combine like terms to simplify the expression to its most basic form.

Like Terms

In algebra, like terms are terms that have the exact same variable parts raised to the same power. Coefficients (the numbers in front of the variables) do not need to be the same.

Here are examples of like terms:

• \(3x\) and \(7x\) are like terms because they both have the variable x raised to the power of one.
• \(5x^2\) and \(-2x^2\) are like terms since they both feature x squared.

In the exercise given,
\(6x^2, 4x^2,\)and \(4x^2\)
are like terms because they all contain the variable x raised to the second power. Similarly,
\(-12x, -3x,\)and \(-10x\)
are like terms because they are x to the first power. Combining like terms is a fundamental step in simplifying algebraic expressions, which helps to reduce the expression into a simpler form.

Simplifying Expressions

Simplifying expressions in algebra means to condense an expression into its simplest form. This includes combining like terms, as seen in the exercise. The step-by-step method of simplifying expressions generally involves:

• Identifying and grouping like terms together.
• Comfortably working through any parentheses by distribution or other means necessary.
• Adding or subtracting the coefficients of like terms.
• Writing down the expression in its simplified form, which has fewer terms and is easier to understand.

From the provided exercise,
\(14x^2 - 25x - 5\)
is the simplified expression obtained by

1. grouping the terms containing the same variables and their corresponding exponents
2. combining their coefficients
3. consolidating constant numbers.

This process makes the overall equation more manageable and is essential for further operations such as solving equations or evaluating the expression for given variable values.