Problem 89 - Elementary and Intermediate Algebra

Question

Simplify. $$ 2\left(s^{2}-7\right)-\left(s^{2}-2\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(s^2 - 12\).

Step 1: Distribute the 2 across the parentheses

Take the expression \(2(s^2 - 7)\) and distribute the 2 to each term inside the parentheses. This means multiplying 2 by each term inside: \(2 \times s^2 - 2 \times 7\). This yields \(2s^2 - 14\).

Step 2: Distribute the negative sign across the parentheses

Take the expression \(-\left(s^2 - 2\right)\) and distribute the negative sign to each term inside the parentheses. This means multiplying -1 by each term inside: \(-1 \times s^2 + 1 \times 2\). This yields \(-s^2 + 2\).

Step 3: Combine like terms

Now combine the results from Step 1 and Step 2: \(2s^2 - 14 - s^2 + 2\). Simplify by combining like terms: \(2s^2 - s^2 = s^2\) and \(-14 + 2 = -12\). Therefore, the expression simplifies to \(s^2 - 12\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term and two or more terms inside a set of parentheses. Imagine you have a bag of candies and you want to distribute the candies to several friends. That's similar to distributing a number across terms in parentheses. Here's how it works:

• Multiply the number outside the parentheses by every term inside the parentheses.
• This operation helps break down expressions into simpler pieces, making it easier to solve them.

For example, in the expression \(2(s^2 - 7)\), you need to distribute the 2 to both \(s^2\) and \(-7\). This results in \(2 \times s^2 - 2 \times 7\), which simplifies to \(2s^2 - 14\).

In another case, when we have a negative sign like in \(- (s^2 - 2)\), we distribute \(-1\) to each term, changing the signs: this becomes \(-1 \times s^2 + 1 \times 2\), or \(-s^2 + 2\). This approach of distributing helps manage and reduce complex algebraic expressions to a simpler form.

Combining Like Terms

Combining like terms is a technique used in algebra to simplify expressions and make them easier to manage. Like terms are terms that have the same variable raised to the same power. Here’s what you need to know:

• Look for terms with the same variable and exponent.
• Add or subtract the coefficients of these terms.

In the expression from our original exercise, after distributing we have: \(2s^2 - 14 - s^2 + 2\).

Now, it's time to combine like terms:

• The terms \(2s^2\) and \(-s^2\) are like terms, because they both have \(s^2\). When combined, they give \(s^2\).
• The constant terms \(-14\) and \(+2\) are also combined, resulting in \(-12\).

So, putting it all together: \(s^2 - 12\). Combining like terms is crucial for simplifying complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They represent specific values or operations and are fundamental in learning algebra.

• Variables are symbols, like \(s\) in this exercise, that represent unknown numbers.
• Expressions can involve operations like addition, subtraction, multiplication, and division.

In our exercise, the initial algebraic expression to simplify was \(2(s^2 - 7) - (s^2 - 2)\). This expression includes variables (\(s^2\)), coefficients (numbers multiplying a variable, like 2), and constants (plain numbers, like \(-7\) and \(2\)).

Understanding how to manipulate these expressions using the distributive property and combining like terms is key to simplifying them. Once simplified, expressions become much less cluttered and more manageable, making it easier to work with them for further calculations or solving equations.