Problem 48 - Algebra 2

Question

Simplify. $$ \left(2 x^{2}-3 x+5\right)-\left(3 x^{2}+x-9\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-x^2 - 4x + 14\).

Step 1: Distribute the Negative Sign

We start by distributing the negative sign in front of the second parenthesis to each term inside the parenthesis: \[ -(3x^2 + x - 9) = -3x^2 - x + 9 \] This changes the original expression to: \[ (2x^2 - 3x + 5) - 3x^2 - x + 9 \]

Step 2: Combine like terms

Now, rewrite the expression in a way that groups similar terms together and combine them:- Start by combining the \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\).- Then, combine the \(x\) terms: \(-3x - x = -4x\).- Finally, combine the constant terms: \(5 + 9 = 14\).This gives us the simplified expression: \[ -x^2 - 4x + 14 \]

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one important process is combining like terms. Like terms are terms within an expression that have the same variable raised to the same power. This means the coefficients (the numbers in front of the variables) can be added or subtracted while keeping the variable part the same.

For example, in the expression \(2x^2 - 3x^2\), both terms are like terms because they have the variable \(x\) raised to the same power (2). We only need to focus on the coefficients: \(2\) and \(-3\). Combine them by subtracting: \(2 - 3 = -1\). So, \(2x^2 - 3x^2 = -x^2\).

Always check for terms with the same variables and exponents when simplifying. It doesn't matter what the coefficients are, only that the variables and their powers match.

The same concept applies to terms without variables, such as constants. If two constants like \(5\) and \(9\) appear in an expression, you can simply add them together to get \(14\).

Combining like terms is crucial in simplifying expressions because it reduces the complexity, making it easier to understand or work with.

Distributive Property

The distributive property is a powerful tool in algebra. It allows us to multiply a single term by the terms within parentheses. This property can be used not just for multiplication, but also for distributing a negative sign.

In our original exercise, the expression \(-(3x^2 + x - 9)\) utilizes the distributive property to change the sign of each term inside the parentheses. This involves multiplying each term by \(-1\). Applying this, the terms in the parentheses change as follows:

• \(3x^2\) becomes \(-3x^2\)
• \(x\) becomes \(-x\)
• \(-9\) becomes \(+9\)

This effectively changes the whole expression to \( -3x^2 - x + 9 \).

Remember, when you have a negative sign in front of parentheses, every term inside must be multiplied by \(-1\). This step is crucial because missing it can lead to errors in solving algebraic expressions.

Mastering the distributive property helps in simplifying expressions and solving equations efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (like \(+\), \(-\), \(*\), \(/\)). They represent quantities and can be simple or quite complex depending on the situation.

In the expression from the exercise, \((2x^2 - 3x + 5) - (3x^2 + x - 9)\), each part represents something specific:

• \(2x^2\), \(-3x^2\) are terms with \(x^2\). These terms indicate the variable \(x\) squared.
• \(-3x\), \(-x\) consist of terms with the variable \(x\) raised to the power of one.
• \(5\) and \(-9\) are constant terms, without variables.

Simplifying algebraic expressions like this is a key step before solving equations or further manipulation. We use operations such as addition, subtraction, and the distributive property to reduce the expression, making it easier to handle.

Understanding algebraic expressions and the rules to manipulate them forms the foundation for solving complex algebra problems. As you practice, these manipulations become second nature, allowing you to confidently tackle diverse mathematical challenges.