Problem 44 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(2 x^{2}+7 x-9\right)+\left(x^{2}-x+10\right)-\left(3 x^{2}-30\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x + 31\).

Step 1: Remove Parentheses

First, we need to remove all the parentheses from the expression. This involves recognizing that addition and subtraction of expressions can be done by directly operating on the terms inside the parentheses. The expression is written as: \[ 2x^2 + 7x - 9 + x^2 - x + 10 - 3x^2 + 30 \]

Step 2: Rearrange and Combine Like Terms

Once we have removed the parentheses, we can rearrange the terms so that like terms (terms with the same variable and exponent) are together. After rearranging, our expression looks like this:\[ (2x^2 + x^2 - 3x^2) + (7x - x) + (-9 + 10 + 30) \]

Step 3: Simplify the Expression

Now, we combine like terms:- For the \(x^2\) terms: \(2x^2 + x^2 - 3x^2 = 0x^2\)- For the \(x\) terms: \(7x - x = 6x\)- For the constant terms: \(-9 + 10 + 30 = 31\)After combining, the expression reduces to:\[ 0x^2 + 6x + 31 \]

Step 4: Finalize the Expression

Since the coefficient of the \(x^2\) term is zero, we can drop this term from the expression to simplify it further. The expression simplifies to:\[ 6x + 31 \]

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in algebra that helps simplify expressions by grouping terms with the same variable and exponent. Imagine like terms as members of the same team - they need to come together to play effectively.

• For instance, terms like \(2x^2\), \(x^2\), and \(-3x^2\) are like terms because they all have \(x^2\) as their variable with the same exponent 2.
• Similarly, terms with just \( x \), such as \( 7x \) and \(-x\), can also be combined because they belong to the same family.

To combine, simply perform arithmetic operations (addition or subtraction) on the coefficients of the like terms. For example, adding the coefficients of \(x^2\) terms: \(2x^2 + x^2 - 3x^2\) equates to 0 because \(2 + 1 - 3 = 0\).
Notice how this method drastically simplifies expressions, reducing complexity and making further calculations more manageable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest and most efficient form without altering their values. This process makes the expression easier to understand and use.
When simplifying, the goal is to:

• Remove parentheses by applying distributive laws, if necessary.
• Combine like terms, as discussed previously, to reduce the number of terms.

Let's consider the expression: \(2x^2 + 7x - 9 + x^2 - x + 10 - 3x^2 + 30\). By removing parentheses, then rearranging and combining like terms, we simplify it to \(6x + 31\).
Simplification reduces computational errors and makes solving equations or further operations straightforward. It’s similar to cleaning up a messy room to find things faster and work more efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving variables, constants, and operational symbols (like addition and subtraction). They serve as the building blocks for more complex mathematical equations and functions.

• Variables, such as \(x\) in the expression \(2x^2 + 7x - 9\), are symbols used to represent unknown numbers.
• Constants are fixed values without a variable attached, like \(-9\) or \(10\) in our example.

Crafting algebraic expressions allows us to translate real-world problems into mathematical language. For instance, an algebraic expression can represent the cost equation using variables for different elements influencing the total cost.
Understanding the components of an algebraic expression is key to manipulating and simplifying them, enabling you to solve equations and understand functions more thoroughly.