Solving 4(x^2-29)-37=747: A Math Guide

Hey math whizzes and curious minds! Today, we're diving deep into a super interesting algebra problem: solving the equation $4\left(x^2-29\right)-37=747$. Now, I know sometimes equations can look a bit intimidating, especially when they involve parentheses and squared terms, but trust me, guys, with a step-by-step approach, this one is totally manageable and, dare I say, even fun to unravel. We're going to break it down piece by piece, making sure every step is clear, so by the end of this, you'll feel confident tackling similar problems. Think of it like solving a puzzle; each number and operation is a clue leading us to the final answer.

Our main goal here is to isolate the variable 'x'. That means we want to get 'x' all by itself on one side of the equals sign. To do this, we'll use the inverse operations. Remember those? They're like the opposite actions that undo each other. For example, the opposite of addition is subtraction, and the opposite of multiplication is division. We'll be using these trusty tools to peel away the numbers surrounding $x^2$ until $x$ is finally free. So, grab your thinking caps, maybe a notepad and pen, and let's get started on this mathematical adventure together. We're aiming to find the value(s) of 'x' that make this equation true. It’s all about careful manipulation and understanding the order of operations in reverse. Ready? Let's go!

Step 1: Simplify the Equation by Isolating the Parenthetical Term

Alright team, the first major step in solving $4\left(x^2-29\right)-37=747$ is to start simplifying the equation. Our current equation has several parts: a term multiplied by a parenthesis, a number being subtracted from that, and then the result on the other side of the equals sign. Our immediate target is to get the part with the parentheses, $4\left(x^2-29\right)$, by itself. Right now, we have that '-37' hanging out there. To get rid of it, we need to perform the inverse operation. Since 37 is being subtracted, we're going to add 37 to both sides of the equation. This is super important, guys, because whatever you do to one side of an equation, you must do to the other side to keep it balanced. It's like a perfectly calibrated scale; you can't just add weight to one side without affecting the balance. So, let's add 37 to both sides:

$4\left(x^2-29\right)-37 + 37 = 747 + 37$

When we do this, the '-37' and '+37' on the left side cancel each other out, leaving us with:

$4\left(x^2-29\right) = 784$

See? We've already made some great progress! The equation is looking a bit cleaner. Now, the term $4\left(x^2-29\right)$ is isolated. This means it's standing alone on one side of the equals sign. This is exactly what we wanted before we tackle what's inside the parentheses. Remember, the goal is to get 'x' by itself, and we're working from the outside in. This step of adding 37 is a classic move in algebra to simplify equations that have constants being added or subtracted outside of grouped terms. It sets us up perfectly for the next stage of our solving journey. Keep up the awesome work!

Step 2: Remove the Multiplication Factor

Okay, fantastic work on Step 1, everyone! Now that we have $4\left(x^2-29\right) = 784$, our next mission is to deal with that '4' that's multiplying the parentheses. Remember, in algebra, when a number is right next to a parenthesis like this, it means multiplication. Our goal is still to get $x^2$ by itself, and that '4' is standing in our way. To undo multiplication, we use its inverse operation: division. So, we're going to divide both sides of the equation by 4. This is crucial for isolating the term inside the parentheses. Remember our scale analogy? Dividing both sides keeps everything perfectly balanced.

Let's do the division:

$\frac{4\left(x^2-29\right)}{4} = \frac{784}{4}$

On the left side, the '4' in the numerator and the '4' in the denominator cancel each other out, simplifying the equation beautifully. On the right side, we need to perform the division $784 \div 4$. If you need to, grab a calculator or do the long division: $784 \div 4 = 196$. So, our equation now becomes:

$x^2-29 = 196$

Boom! We've successfully removed the multiplication factor. The expression inside the parentheses, $x^2-29$, is now completely isolated. This is a huge win because we're getting closer and closer to having just $x^2$ on its own. This step is fundamental in solving equations where a group of terms is multiplied by a coefficient. By dividing both sides, we essentially 'unwrap' the equation, getting us closer to the core variable. You guys are doing great! Stay focused, we're on the home stretch.

Step 3: Isolate the Squared Term

Alright, brilliant job so far, team! We've simplified the equation down to $x^2-29 = 196$. Now, we need to get $x^2$ completely by itself. Currently, we have '-29' attached to it. To isolate $x^2$, we need to undo the subtraction of 29. What's the opposite of subtracting 29, guys? That's right – adding 29! We'll add 29 to both sides of the equation to maintain that all-important balance.

Let's add 29 to both sides:

$x^2-29 + 29 = 196 + 29$

On the left side, the '-29' and '+29' cancel each other out, leaving us with just $x^2$. On the right side, we perform the addition: $196 + 29 = 225$. So, our equation simplifies further to:

$x^2 = 225$

And there we have it! The term $x^2$ is now isolated. This is a pivotal moment because $x^2$ is just one step away from being $x$. We've successfully unwrapped the equation, dealing with the subtraction outside the parenthesis, the multiplication factor, and finally the subtraction inside. All that's left is to deal with the exponent. Keep that momentum going!

Step 4: Solve for 'x' by Taking the Square Root

We've reached the final, exciting step in solving $4\left(x^2-29\right)-37=747$: getting $x$ by itself from $x^2 = 225$. To undo the operation of squaring a number (raising it to the power of 2), we need to perform its inverse operation, which is taking the square root. Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because $3 \times 3 = 9$. So, we will take the square root of both sides of the equation:

$\sqrt{x^2} = \sqrt{225}$

On the left side, the square root of $x^2$ is simply $x$. However, and this is a really important point in algebra, when you take the square root of both sides of an equation to solve for a variable, there are two possible solutions: a positive one and a negative one. This is because both a positive number multiplied by itself and its negative counterpart multiplied by itself result in a positive number. For instance, $15 \times 15 = 225$, and also $(-15) \times (-15) = 225$. Therefore, $\sqrt{225}$ is not just 15, but ±15 (read as 'plus or minus 15').

So, our solutions for $x$ are:

$x = 15$ and $x = -15$

These are the two values of 'x' that satisfy the original equation. We've successfully navigated through all the steps, from simplifying to isolating and finally solving for our variable. High fives all around, guys! You've conquered this algebraic challenge by systematically applying inverse operations and understanding the properties of equations.

Conclusion: The Solution and What We Learned

So there you have it, folks! After carefully working through each step, we've discovered that the equation $4\left(x^2-29\right)-37=747$ has two solutions: $x = 15$ and $x = -15$. We successfully solved the equation by using the fundamental principles of algebra. We started by isolating the term containing the variable, using inverse operations like addition to remove constants and division to remove coefficients. Then, we isolated the squared term $x^2$, and finally, we took the square root of both sides, remembering to account for both positive and negative possibilities. This journey highlights the power of inverse operations in simplifying and solving algebraic equations. It's like a structured way of untangling a complex problem. Each step logically follows from the previous one, leading us directly to the answer. Remember these steps, and you'll be well-equipped to tackle a wide variety of similar algebraic equations. Keep practicing, keep exploring, and never shy away from a challenging equation – you've got this!