Simplifying Algebraic Expressions Finding Equivalent Expressions

Hey guys! Today, we're diving into a common type of math problem: simplifying algebraic expressions. We'll break down a specific example step-by-step, ensuring you understand the process thoroughly. This will not only help you ace your exams but also build a solid foundation for more advanced math concepts. So, let’s get started and make math a little less intimidating!

The Problem: Unveiling the Equivalent Expression

Our mission, should we choose to accept it, is to find the expression equivalent to this beauty: $3x^2 + 5x - 7(x^2 + 4)$. We have four potential answers:

A. $-4x^2 + 5x - 4$ B. $-4x^2 + 5x - 28$ C. $x^2 + 4$ D. $x^2 + 28$

Which one is the real deal? Let's find out!

Step 1: Mastering the Distributive Property

The golden rule here is the distributive property. This basically means we need to multiply the term outside the parentheses (in this case, -7) by each term inside the parentheses ($x^2$ and +4). Think of it like sharing the -7 with everyone inside the house!

So, we have:

$-7 * x^2 = -7x^2$

and

$-7 * 4 = -28$

Now, let's rewrite the original expression with this distribution applied:

$3x^2 + 5x - 7(x^2 + 4)$ becomes $3x^2 + 5x - 7x^2 - 28$

See how we've expanded the expression? We're one step closer to the simplified form.

Step 2: Combining Like Terms – It’s a Group Effort!

Next up, we need to combine like terms. This is like sorting your socks – you put all the pairs together! In our expression, "like terms" are those with the same variable raised to the same power. We're looking for terms that have the same variable and exponent. Constants (just numbers) are also like terms.

In our expanded expression, we have:

• $3x^2$ and $-7x^2$ (both have $x^2$)
• $5x$ (this one's a loner for now, as it's the only term with just $x$)
• $-28$ (a constant, hanging out by itself)

Let's combine the $x^2$ terms:

$3x^2 - 7x^2 = -4x^2$

Now, let's rewrite the entire expression with the combined terms:

$-4x^2 + 5x - 28$

Voila! We've simplified the expression.

Step 3: Spotting the Answer

Now, let's compare our simplified expression, $-4x^2 + 5x - 28$, with the answer choices we were given:

A. $-4x^2 + 5x - 4$ B. $-4x^2 + 5x - 28$ C. $x^2 + 4$ D. $x^2 + 28$

Ding ding ding! Answer choice B is a perfect match. We've found our equivalent expression!

Why Other Options Are Incorrect

It's always good to understand why the other options are wrong. This helps solidify your understanding of the concepts.

• A. $-4x^2 + 5x - 4$: This option seems to have made a mistake in the distribution or combining the constant terms. The -28 from the original distribution was likely incorrectly calculated.
• C. $x^2 + 4$: This choice completely ignores the initial distribution and combining like terms. It's a far cry from the correct answer.
• D. $x^2 + 28$: Similar to option C, this one misses the crucial steps of distribution and combining like terms. It's as if the problem was approached with a completely different strategy.

Key Concepts: Your Toolkit for Success

Let's recap the key concepts we used to solve this problem. These are your tools for tackling similar expressions:

1. Distributive Property: This is the cornerstone of simplifying expressions with parentheses. Remember to multiply the term outside the parentheses by every term inside.
2. Combining Like Terms: Group together terms with the same variable and exponent. This simplifies the expression and makes it easier to work with.
3. Order of Operations (PEMDAS/BODMAS): While not explicitly used in this problem, remember the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) for more complex expressions.

Real-World Applications: Where Algebra Comes Alive

You might be thinking, "Okay, this is great for math class, but when will I ever use this in real life?" Well, simplifying algebraic expressions pops up in many unexpected places:

• Engineering: Engineers use algebraic expressions to model and analyze systems, from circuits to bridges.
• Computer Science: Simplifying expressions is crucial in optimizing code and algorithms.
• Finance: Financial analysts use algebraic expressions to calculate investments, interest rates, and other financial metrics.
• Everyday Life: Even balancing your budget involves simplifying expressions! You might not realize it, but you're using these skills all the time.

Practice Makes Perfect: Sharpening Your Skills

The best way to master simplifying expressions is through practice. Here are a few tips:

• Work through examples: Start with simple expressions and gradually increase the complexity.
• Identify common mistakes: Pay attention to areas where you tend to make errors, such as distributing negative signs or combining unlike terms.
• Use online resources: There are tons of websites and apps that offer practice problems and step-by-step solutions.
• Ask for help: Don't be afraid to ask your teacher, classmates, or online communities for assistance.

Conclusion: You've Got This!

Simplifying algebraic expressions might seem daunting at first, but with a systematic approach and a little practice, you'll become a pro in no time. Remember the distributive property, combine like terms, and don't forget to double-check your work. You've got this!

So, the answer to our original question, "Which expression is equivalent to $3x^2 + 5x - 7(x^2 + 4)$?" is B. $-4x^2 + 5x - 28$. Keep practicing, and you'll be simplifying expressions like a math whiz!

Remember, guys, math is like a puzzle – challenging but ultimately rewarding when you crack the code. Keep learning, keep practicing, and keep shining!