Problem 54 - Algebra for College Students

Question

Perform the indicated operations. $$\left(-6 x^{2}+2 x+5\right)-\left(4 x^{2}+4 x-1\right)+\left(7 x^{2}+4\right)$$

Step-by-Step Solution

Answer

\(-3x^2 - 2x + 10\)

Step 1: Remove Parentheses

First, we need to remove the parentheses and distribute any negative signs across the terms inside the parentheses: \(-6x^2 + 2x + 5 - 4x^2 - 4x + 1 + 7x^2 + 4\).

Step 2: Combine Like Terms

Now, we group and combine like terms. We have:- Quadratic terms: \(-6x^2 - 4x^2 + 7x^2\)- Linear terms: \(2x - 4x\)- Constant terms: \(5 + 1 + 4\).

Step 3: Simplify Quadratic Terms

Combine the coefficients of the quadratic terms: \(-6x^2 - 4x^2 + 7x^2 = (-6 - 4 + 7)x^2 = -3x^2\).

Step 4: Simplify Linear Terms

Combine the coefficients of the linear terms: \(2x - 4x = (-2)x\).

Step 5: Simplify Constant Terms

Combine the constant terms: \(5 + 1 + 4 = 10\).

Step 6: Write the Simplified Expression

Combine all the simplified parts: \(-3x^2 - 2x + 10\).

Key Concepts

Combining Like Terms

In algebra, combining like terms makes expressions easier to work with. When we talk about "like terms," we're referring to terms that have the same variables raised to the same power. It's like grouping apples with apples and oranges with oranges. In the exercise, we had terms like \(-6x^2\), \(2x\), and constants like \(5\).
To combine them, you should:

• Identify terms with the same variable and exponent
• Combine their coefficients by adding or subtracting

For instance, \(-6x^2\), \(-4x^2\), and \(7x^2\) are like terms because they all have \(x^2\) as part of them. So, they can be combined by just working with the numbers (coefficients) in front of the \(x^2\), giving us \(-3x^2\) after simplifying.
This makes it much simpler to handle or solve equations where these expressions might be used.

Simplifying Expressions

Simplifying expressions is about making them too easy to work with and look nicer. After combining like terms, the expression can usually be written more clearly and concisely. Think of it as tidying up a messy room, so you can find things more easily.
For example, from the exercise, the original expression is quite a handful, but once simplify it step by step, we arrive at the tidy form \(-3x^2 - 2x + 10\). This involves taking an expression like \(2x - 4x\) and simplifying it to \(-2x\), by combining the coefficients.
The process involves:

• Collecting like terms
• Performing arithmetic operations to reduce them

The goal is to end up with the simplest version of your expression, one that retains all its information but none of the clutter.

Distributive Property

The distributive property is a rule that helps us multiply a term across a bracketed term. It's fundamental in removing parentheses and simplifying expressions. The property states that \(a(b + c) = ab + ac\). When dealing with subtraction, it's the same idea: distribute the negative sign.
In our exercise, the distributive property was used when we encountered this part: \(-\left(4x^2 + 4x - 1\right)\). We distributed the \(-1\) across the terms inside the parentheses to get \(-4x^2 - 4x + 1\). This change in sign for each term inside the parentheses is key.
Here’s what to remember when using this property:

• Apply the multiplier (or sign) to each term inside the parentheses
• Ensure each multiplication or distribution respects the operation (negative/positive signs)

Using the distributive property correctly helps keep your work accurate and prevent errors in algebraic manipulations.