Find The Expression Equivalent To (3x^2 - 7) A Step-by-Step Guide

Hey guys! Today, we're diving deep into a super common algebra question: finding equivalent expressions. Specifically, we're going to figure out which expression is the same as $(3x^2 - 7)$. It might seem a bit tricky at first, but don't worry, we'll break it down step-by-step so you'll be a pro in no time. Think of this as a puzzle – we're given a target expression, and we need to find which of the other expressions is just a different way of writing the same thing. So, let's jump right in and solve this thing together!

Understanding Equivalent Expressions

Before we even look at the options, let's quickly recap what equivalent expressions actually mean. In simple terms, equivalent expressions are algebraic expressions that look different but have the same value for any given value of the variable (in our case, 'x'). Imagine them as different roads leading to the same destination. They might have different turns and scenery, but they ultimately get you to the same place. To determine if expressions are equivalent, we need to simplify them and see if they end up being identical. This usually involves combining like terms – terms with the same variable and exponent – and using the distributive property to get rid of any parentheses. So, if we simplify an expression and it matches our target expression of $(3x^2 - 7)$, then bingo! We've found our match. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems later on. Remember, practice makes perfect, so don't be afraid to work through lots of examples!

Evaluating the Options

Now, let's get our hands dirty and evaluate the given options to see which one matches $(3x^2 - 7)$. We have four potential contenders, and our mission is to simplify each of them and compare the result with our target expression. This is where the fun begins! We'll take each option one by one, apply the rules of algebra, and whittle them down to their simplest form. Think of it like a detective solving a mystery – we're gathering clues (simplifying the expressions) to uncover the truth (which expression is equivalent). Here’s how we will approach each option:

1. Distribute: If there are parentheses with a minus sign in front, we need to distribute that negative sign to each term inside the parentheses.
2. Combine Like Terms: Look for terms with the same variable and exponent (like $x^2$ terms) and constant terms (numbers without variables). Add or subtract their coefficients.
3. Compare: Once we've simplified the expression as much as possible, we'll compare it to our target, $(3x^2 - 7)$. If they match, we've found our winner!

Let's put on our algebraic hats and get started!

Option 1: $(2x^2 - 11) - (x^2 + 4)$

Okay, let's tackle the first option: $(2x^2 - 11) - (x^2 + 4)$. Remember our strategy? First, we distribute the negative sign in front of the second set of parentheses. This is super important because it changes the signs of the terms inside. So, the expression becomes:

$2x^2 - 11 - x^2 - 4$

Now comes the fun part: combining like terms. We've got two $x^2$ terms ($2x^2$ and $-x^2$) and two constant terms (-11 and -4). Let's group them together:

$(2x^2 - x^2) + (-11 - 4)$

Now we perform the operations. $2x^2 - x^2$ is simply $x^2$ (think of it as 2 'somethings' minus 1 'something' leaves you with 1 'something'). And -11 - 4 is -15. So, our simplified expression is:

$x^2 - 15$

Now, let's compare this to our target expression, $(3x^2 - 7)$. Are they the same? Nope! $x^2 - 15$ has a coefficient of 1 in front of the $x^2$ term and a constant term of -15, while our target has a coefficient of 3 and a constant term of -7. So, Option 1 is not the equivalent expression we're looking for. But hey, that's okay! We've still got three more options to check. Let's move on to the next one and keep our algebraic brains firing!

Option 2: $(5x^2 - 6) - (2x^2 - 1)$

Alright, let's jump into Option 2: $(5x^2 - 6) - (2x^2 - 1)$. Just like before, our first step is to distribute the negative sign in front of the second set of parentheses. Remember, this means we change the sign of every term inside those parentheses. So, we get:

$5x^2 - 6 - 2x^2 + 1$

Notice how the $-2x^2$ term is now negative, and the -1 term has become +1. This is crucial for getting the right answer! Now, let's gather our like terms. We have $5x^2$ and $-2x^2$ as our $x^2$ terms, and -6 and +1 as our constant terms. Let's group them up:

$(5x^2 - 2x^2) + (-6 + 1)$

Time to simplify! $5x^2 - 2x^2$ gives us $3x^2$ (5 'somethings' minus 2 'somethings' equals 3 'somethings'). And -6 + 1 equals -5. So, our simplified expression is:

$3x^2 - 5$

Now, the moment of truth! Let's compare this to our target expression, $(3x^2 - 7)$. Are they a match? Not quite! We have the same $3x^2$ term, which is a good sign, but our constant term is -5, while the target expression has -7. So, Option 2 is also not the equivalent expression we're searching for. Don't lose heart, though! We're getting closer, and we've learned something from each option we've tackled. Onward to Option 3!

Option 3: $(10x^2 - 4) - (7x^2 + 3)$

Okay, let's keep the momentum going with Option 3: $(10x^2 - 4) - (7x^2 + 3)$. You know the drill by now – our first step is to distribute that sneaky negative sign lurking in front of the second set of parentheses. This means we flip the signs of the terms inside: the $7x^2$ becomes $-7x^2$, and the +3 becomes -3. So, the expression transforms into:

$10x^2 - 4 - 7x^2 - 3$

Fantastic! Now comes the satisfying part – grouping our like terms together. We've got $10x^2$ and $-7x^2$ as our $x^2$ buddies, and -4 and -3 hanging out as our constant chums. Let's bring them together for a little algebraic party:

$(10x^2 - 7x^2) + (-4 - 3)$

Time for the simplification magic! $10x^2 - 7x^2$ gives us $3x^2$ (ten 'somethings' minus seven 'somethings' leaves three 'somethings'). And -4 - 3 combines to give us -7. So, our simplified expression is:

$3x^2 - 7$

Drumroll, please! Let's compare this to our target expression, $(3x^2 - 7)$. Ding ding ding! We have a winner! $3x^2 - 7$ is exactly the same as our target expression. We've found our equivalent expression! But just to be absolutely sure, and for the sake of thoroughness, let's quickly check Option 4 as well.

Option 4: $(15x^2 + 8) - (18x^2 + 1)$

Alright, let's wrap things up by examining Option 4: $(15x^2 + 8) - (18x^2 + 1)$. By now, we're practically pros at this! First things first, we distribute the negative sign chilling in front of the second set of parentheses. This changes the $18x^2$ to $-18x^2$ and the +1 to -1, giving us:

$15x^2 + 8 - 18x^2 - 1$

Now, let's gather our like terms for a little algebraic get-together. We've got $15x^2$ and $-18x^2$ representing our $x^2$ crew, and +8 and -1 as our constant companions. Let's group them up:

$(15x^2 - 18x^2) + (8 - 1)$

Time for some simplification action! $15x^2 - 18x^2$ results in $-3x^2$ (fifteen 'somethings' minus eighteen 'somethings' leaves negative three 'somethings'). And 8 - 1 equals 7. So, our simplified expression is:

$-3x^2 + 7$

Let's compare this to our target expression, $(3x^2 - 7)$. Definitely not a match! We have a negative $x^2$ term and a positive constant term, while our target has a positive $x^2$ term and a negative constant term. So, Option 4 is not the equivalent expression we were seeking. And that confirms it – Option 3 is indeed our winner!

Conclusion: The Equivalent Expression

So, after carefully evaluating all four options, we've found that the expression equivalent to $(3x^2 - 7)$ is:

$(10x^2 - 4) - (7x^2 + 3)$

We arrived at this answer by systematically simplifying each expression and comparing it to our target. Remember, the key steps are distributing the negative sign (if there is one) and combining like terms. This is a fundamental skill in algebra, and mastering it will help you tackle all sorts of problems. Keep practicing, and you'll be simplifying expressions like a pro in no time! Great job working through this problem with me, guys! Keep up the awesome work!

Key Takeaways

• Equivalent expressions have the same value for all values of the variable.
• To find equivalent expressions, simplify each expression.
• Distribute negative signs carefully.
• Combine like terms (terms with the same variable and exponent).
• Compare the simplified expressions to the target expression.

By following these steps, you'll be able to confidently identify equivalent expressions in any algebraic problem.