Make X The Subject Solve A = Bx √(x² - 9) Algebra Guide
Making a variable the subject of a formula is a crucial skill in algebra and mathematics. In this article, we'll dive deep into how to manipulate equations to isolate a specific variable. Specifically, we'll tackle the equation a = bx / √(x² - 9), where our goal is to make x the subject. This means we want to rearrange the equation so that it reads x = [some expression involving a and b]. This process involves several algebraic techniques, including squaring both sides, dealing with fractions, and rearranging terms. It’s like solving a puzzle where each step brings us closer to the final answer. So, let's put on our algebraic thinking caps and get started!
Understanding the Basics
Before we jump into the nitty-gritty of rearranging the formula a = bx / √(x² - 9), let's quickly revisit some fundamental algebraic principles. Think of these principles as the building blocks we'll use to construct our solution. First up is the concept of inverse operations. Every mathematical operation has an inverse that undoes it. For example, the inverse of addition is subtraction, and the inverse of multiplication is division. Similarly, the inverse of squaring a number is taking its square root, and vice versa. When we're rearranging an equation, we use inverse operations to isolate the variable we're interested in. Imagine you're unwrapping a gift – you need to undo each layer one by one to get to the present inside. In our case, the present is x, and the layers are the various operations and terms surrounding it.
Next, we need to remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side. This ensures that the equation remains balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle is crucial for maintaining the equality throughout our rearrangement process. We'll be using this rule extensively as we square both sides, multiply, divide, and rearrange terms. It's the foundation upon which all algebraic manipulations are built. Keeping this principle in mind will help us avoid common mistakes and ensure we arrive at the correct solution.
Finally, let's not forget about the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). While PEMDAS is typically used for simplifying expressions, it also guides us in reverse when rearranging equations. We often need to undo operations in the reverse order of PEMDAS, starting with addition and subtraction, then moving on to multiplication and division, and finally dealing with exponents and roots. This order helps us to systematically peel away the layers surrounding x and bring it closer to being the subject of the formula. So, with these basic principles in mind, we're well-equipped to tackle the challenge of making x the subject of a = bx / √(x² - 9). Let's get to it, guys!
Step-by-Step Solution
Okay, guys, let's get down to business and solve this equation step-by-step! Our mission, should we choose to accept it (and we do!), is to make x the subject of the formula a = bx / √(x² - 9). Remember, this means we want to isolate x on one side of the equation. So, grab your pencils, and let's dive in!
Step 1: Squaring Both Sides
The first thing we need to deal with is that pesky square root in the denominator. Square roots can be a bit of a headache to work with directly, so our initial move is to get rid of it. How do we do that? By squaring both sides of the equation! Squaring is the inverse operation of taking a square root, so it neatly cancels out the square root on the right-hand side. When we square both sides of a = bx / √(x² - 9), we get:
a² = (bx)² / (x² - 9)
Notice how the square root on the denominator magically disappeared? That's the power of inverse operations in action! Squaring both sides not only eliminates the square root but also sets us up for the next steps in isolating x. It's like clearing the first hurdle in a race – we've made significant progress, but there's still more to go. The left side becomes a², and the right side simplifies because the square applies to both bx (becoming b²x²) and the square root, effectively cancelling it out and leaving us with (x² - 9) in the denominator. This transformation is a key step because it removes a major obstacle and brings us closer to our goal. So, let's keep the momentum going and move on to the next step!
Step 2: Multiplying Both Sides by (x² - 9)
Now that we've squared both sides and gotten rid of the square root, we have a² = b²x² / (x² - 9). The next thing we want to tackle is the fraction. Fractions can be a bit cumbersome to work with, so let's get rid of the denominator. To do this, we'll multiply both sides of the equation by (x² - 9). Remember the golden rule of algebra – what we do to one side, we must do to the other to maintain balance. Multiplying both sides by (x² - 9) gives us:
a²(x² - 9) = b²x²
See how the (x² - 9) on the right side cancels out with the (x² - 9) we multiplied by? This leaves us with a much cleaner equation to work with. It's like simplifying a complex recipe by removing unnecessary ingredients. The left side now has a² multiplied by the quantity (x² - 9), and the right side is simply b²x². This step is crucial because it eliminates the fraction and allows us to start rearranging the terms more easily. By getting rid of the denominator, we've opened up new possibilities for isolating x. So, let's keep pushing forward and see what the next step reveals!
Step 3: Expanding and Rearranging
Alright, guys, we're making great progress! We've squared both sides and multiplied to get rid of the fraction, and now we have a²(x² - 9) = b²x². The next step is to expand the left side of the equation and then rearrange the terms so that all the x² terms are on one side and the constants are on the other. This is a classic algebraic technique that helps us group like terms together, making it easier to isolate x. First, let's expand the left side by distributing the a²:
a²x² - 9a² = b²x²
Now we have a more spread-out equation, which is perfect for rearranging. Our goal is to get all the x² terms on one side, so let's subtract a²x² from both sides:
-9a² = b²x² - a²x²
We've successfully moved the x² term to the right side. Now, let's factor out the x² from the right side. This is like reverse distribution, where we pull out the common factor:
-9a² = x²(b² - a²)
By expanding and rearranging, we've transformed the equation into a form where x² is almost isolated. It's like organizing your toolbox before starting a project – having everything in its place makes the job much easier. We're one step closer to making x the subject of the formula. So, let's keep up the momentum and see what the next step entails!
Step 4: Isolating x²
We're on the home stretch now, guys! We've expanded and rearranged the equation, and we're currently sitting pretty with -9a² = x²(b² - a²). Our next mission, should we choose to accept it (of course, we do!), is to isolate x². This means we need to get x² all by itself on one side of the equation. Looking at our equation, we see that x² is being multiplied by (b² - a²). So, what's the inverse operation of multiplication? You guessed it – division! To isolate x², we'll divide both sides of the equation by (b² - a²). Remember the golden rule – keep the equation balanced!
Dividing both sides by (b² - a²) gives us:
-9a² / (b² - a²) = x²
Hooray! We've successfully isolated x². It's like reaching the summit of a mountain after a long climb – we can see the finish line from here. But before we plant our flag, we need to take one more step to get x all by itself. Notice that x is still squared. We need to undo that square to finally make x the subject of the formula. So, let's move on to the final step and conquer this equation!
Step 5: Taking the Square Root
We've made it to the final showdown, guys! We've navigated through squaring, multiplying, expanding, rearranging, and dividing, and now we have -9a² / (b² - a²) = x². The last step in our quest to make x the subject of the formula is to get rid of that pesky square. We know that the inverse operation of squaring is taking the square root. So, let's take the square root of both sides of the equation.
Taking the square root of both sides gives us:
x = ±√(-9a² / (b² - a²))
And there you have it! We've successfully made x the subject of the formula. It's like unlocking the final level in a video game – we've overcome all the challenges and emerged victorious. Notice the ± sign in front of the square root. This is because when we take the square root, we need to consider both the positive and negative roots. A positive number squared gives a positive result, but so does a negative number squared. So, we need to include both possibilities to be mathematically complete.
Before we declare mission accomplished, let's just tidy things up a little. We can simplify the expression inside the square root by multiplying the numerator and denominator by -1 to get rid of the negative sign in the numerator:
x = ±√(9a² / (a² - b²))
This is a slightly cleaner way to express our final answer. Now, we can confidently say that we've made x the subject of the formula a = bx / √(x² - 9). Give yourselves a pat on the back, guys! You've conquered a challenging algebraic problem with skill and perseverance.
Final Answer
After a journey through squaring, multiplying, expanding, rearranging, dividing, and finally taking the square root, we've successfully made x the subject of the formula a = bx / √(x² - 9). Our final answer is:
x = ±√(9a² / (a² - b²))
This means that x is equal to either the positive or the negative square root of (9a² / (a² - b²)). The ± sign is crucial because it reminds us that there are two possible solutions for x when we take the square root. It's like finding two keys that unlock the same door – both are valid solutions.
To recap, we started with the original formula and systematically applied algebraic techniques to isolate x. We squared both sides to get rid of the square root, multiplied to eliminate the fraction, expanded and rearranged terms to group like terms together, divided to isolate x², and finally took the square root to solve for x. Each step was a crucial piece of the puzzle, and by following them carefully, we arrived at our final answer.
This process demonstrates the power of algebraic manipulation and the importance of understanding inverse operations. It's like learning a new language – once you grasp the grammar and vocabulary, you can express yourself in countless ways. In this case, we've learned how to rearrange formulas to solve for different variables, a skill that's invaluable in mathematics and many other fields.
So, there you have it, guys! We've not only found the answer but also walked through the reasoning and steps involved. Remember, the journey is just as important as the destination. By understanding the process, you'll be better equipped to tackle similar problems in the future. Keep practicing, and you'll become a master of algebraic manipulation!
Repair Input Keyword
Okay, let's talk about the keyword we started with: "Make the subject of the formula ". While the original question is clear to those familiar with mathematical notation, we can make it even more accessible and easier to understand for everyone. Think of it as translating from math-speak to plain English. So, let's repair this keyword to make it super clear and user-friendly.
A slightly improved version of the keyword could be: "How to make x the subject of the formula a = bx / √(x² - 9)". This version is more conversational and directly asks a question, which is what many people type into search engines. It also retains all the essential mathematical elements, ensuring that we're still addressing the original problem. It's like adding a friendly introduction to a formal request – it makes it more approachable and less intimidating.
Another way to phrase the keyword is: "Solve for x in the equation a = bx / √(x² - 9)". This version uses the phrase "solve for x," which is a common term in algebra. It's like using a widely understood shorthand – it gets the message across quickly and efficiently. It also replaces the phrase "make x the subject" with a more direct instruction, which can be helpful for those who are looking for a straightforward solution.
By repairing the input keyword, we're essentially optimizing it for search engines and human understanding. It's like fine-tuning a radio signal to get the clearest reception – we want to make sure that anyone searching for this type of problem can easily find our solution. A well-crafted keyword is like a beacon, guiding people to the information they need. So, by making our keywords clear, concise, and user-friendly, we're making math a little less daunting for everyone. Remember, guys, clear communication is key, whether we're talking about algebra or anything else!
SEO Title
Okay, let's craft an SEO-friendly title for this article! The goal here is to create a title that not only accurately reflects the content but also attracts readers and ranks well in search engine results. It's like writing a catchy headline for a news story – you want to grab people's attention and make them want to click. So, let's put on our SEO hats and get to work!
Here are a few options we could consider:
- Make x the Subject of the Formula A Comprehensive Guide
- Solve for x in a bx / √(x² - 9) Step-by-Step Solution
- Isolate x Formula Manipulation Explained
But let's go with this title:
Make x the Subject Solve a = bx √(x² - 9) Algebra Guide
This title is concise, includes the main keywords ("Make x the subject," "solve," "algebra"), and directly references the formula we're working with. It's like packing all the essential information into a small suitcase – we're maximizing the impact with a limited number of words. It avoids using a colon (:) which can sometimes cause issues with certain systems and SEO best practices recommend keeping titles concise for better readability and search engine ranking.
The phrase "Make x the subject" is a direct and clear indication of the article's focus. It's like putting a sign on the door that says exactly what's inside. The inclusion of "solve" broadens the appeal and captures those searching for general equation-solving techniques. It's like casting a wider net to catch more fish. The formula itself is included to provide context and ensure that readers know exactly what problem we're addressing. It's like showing a picture of the puzzle we're about to solve.
Finally, the term "Algebra Guide" positions the article as a helpful resource for those studying algebra. It's like labeling a product as "premium" – it suggests high quality and comprehensive information. So, by carefully crafting our SEO title, we're increasing the chances that our article will be discovered and read by those who need it most. Remember, guys, a great title is the first step in sharing your knowledge with the world!
