Find X When F(x) = 0: A Step-by-Step Guide

Hey guys! Let's dive into a classic math problem: finding the value of x that makes a function equal to zero. This is super important in algebra and has tons of real-world applications. We're gonna focus on the equation f(x) = 2√(x + 3) - 6. Our goal is to find the value(s) of x that satisfy f(x) = 0. So, basically, where does the graph of this function cross the x-axis? Let's break it down step-by-step, making sure it's all clear and easy to follow. We'll get there together!

Understanding the Problem: The Basics of Finding Roots

So, what does it actually mean to find x such that f(x) = 0? Well, the value(s) of x that satisfy this condition are called the roots or zeros of the function. Think of it like this: you're looking for the x-values where the function's output (y-value) is zero. Graphically, this represents the points where the function's graph intersects the x-axis. Why is this important, you ask? Because finding roots helps us solve various problems. For example, in physics, we might use this to find the time when a projectile hits the ground (when its height, h(t), is zero). In business, we might use it to find the break-even point of a cost function (when profit, P(x), is zero). It's a fundamental concept! The equation we're working with, f(x) = 2√(x + 3) - 6, is a bit different because it involves a square root. This means we need to pay special attention to the domain of the function (the allowed values of x) to make sure we don't end up with any mathematical nonsense like the square root of a negative number. We'll definitely keep that in mind as we go through the calculations. Now, finding roots can sometimes be straightforward, and sometimes it requires a bit more algebraic manipulation. For our equation, we'll aim for a simple, direct approach. This includes isolating the radical term, squaring both sides (being careful about potential extraneous solutions), and then solving for x. It's all pretty manageable; we just need to be organized and keep track of our steps. So, let's get cracking and find that value of x!

Solving the Equation: Step-by-Step Guide

Alright, let's get our hands dirty and actually solve the equation f(x) = 2√(x + 3) - 6 = 0. We want to isolate x, so let's take it one step at a time, like good little mathematicians. First, we'll tackle the square root, and remember, we need to make sure we're working with valid values of x. This is the whole point of finding solutions; we are going to find a single solution for x. Here's how we will do it:

1. Isolate the Radical: The first thing we need to do is get the square root term by itself on one side of the equation. We do this by adding 6 to both sides. This gives us:

   2√(x + 3) = 6

   We add six on both sides to isolate the radical. At this point, the goal is to simply isolate the radical to start working with a simpler form. Next, we'll take care of the next part of the problem. This means removing the coefficient next to the radical, which is the next logical step.

2. Divide to Simplify: Now, let's divide both sides of the equation by 2. This simplifies things a bit more:

   √(x + 3) = 3

   By dividing both sides by two, we are again simplifying the equation. Now the equation is more manageable, as we only have the radical and constants. We're inching closer to isolating x.

3. Square Both Sides: To get rid of the square root, we square both sides of the equation. Remember, whatever you do to one side, you gotta do to the other. This gives us:

   x + 3 = 9

   When squaring, we get rid of the radical, which gives us a simpler form. The purpose of squaring both sides is to eliminate the square root, which is our goal at this stage. This means getting rid of that pesky radical so we can solve for x. We must remember that squaring can sometimes introduce extraneous solutions, which is something we'll keep in mind. This is crucial, guys!

4. Solve for x: Now it's just a matter of solving for x. Subtract 3 from both sides:

   x = 6

   Simple arithmetic! By subtracting three from both sides of the equation, we isolate x on one side of the equation. Congratulations, we've got an x value!

So, we've got x = 6. Seems pretty straightforward, right? But there's one more very important step before we declare victory, and that is to check our solution. This step is essential to ensure our solution is correct and valid.

Checking the Solution: Verifying Your Answer

Okay, we've found that x = 6. But before we start celebrating, we always need to check our answer. This is super important, especially when dealing with equations that involve square roots. Checking involves plugging the solution back into the original equation and making sure it holds true. We do this to ensure that our calculations were correct and that we don't have any extraneous solutions (solutions that arise from the algebra but don't actually work in the original equation). Let's substitute x = 6 back into f(x) = 2√(x + 3) - 6:

f(6) = 2√(6 + 3) - 6

f(6) = 2√(9) - 6

f(6) = 2 * 3 - 6

f(6) = 6 - 6

f(6) = 0

And there you have it! Our solution, x = 6, satisfies the original equation. This confirms that the value we found is indeed the root of the function. We have found the correct answer! It’s always a good idea to get in the habit of checking your solutions to avoid any mistakes. Not only does it ensure the accuracy of your answer, but it also helps reinforce your understanding of the underlying concepts. Yay!

Conclusion: The Root Revealed

Alright, we've made it! We've successfully found the value of x that makes f(x) = 2√(x + 3) - 6 equal to zero. By following a step-by-step approach, including isolating the radical, squaring both sides, and solving for x, we determined that x = 6. But beyond just finding the answer, we also emphasized the importance of checking your solution. Plugging our answer back into the original equation ensured our result was valid and correct. So, when you're tackling similar problems, remember the importance of methodical steps and diligent checking! This process is applicable to all root-finding problems. Knowing this will give you a huge advantage! Math problems can be intimidating at times. But by breaking them down into smaller, manageable parts, we can solve them effectively and confidently. Keep practicing, keep exploring, and never stop asking “why.” You got this, guys! Now go forth and conquer those equations! We hope this article was helpful. Thanks for reading!