Solving The Equation √1-3x = X + 3 A Step-by-Step Guide
Hey guys! Ever stumbled upon a seemingly simple equation that turned out to be a bit of a puzzle? Today, we're diving deep into solving the equation √1-3x = x + 3. This might look straightforward, but there are a few sneaky steps we need to take to make sure we arrive at the correct solution. We'll break it down piece by piece, so you'll not only understand how to solve it but also why we take each step. This isn't just about getting the answer; it's about mastering the process. So, let's get started and conquer this mathematical challenge together!
Understanding the Equation
Before we jump into the solution, let's take a moment to understand what we're dealing with. The equation √1-3x = x + 3 involves a square root, which adds a layer of complexity. Remember, the square root of a number is only defined for non-negative values. This means the expression inside the square root, 1-3x, must be greater than or equal to zero. This gives us our first constraint: 1-3x ≥ 0. Solving this inequality, we get x ≤ 1/3. This is crucial because any solution we find later must satisfy this condition. If it doesn't, it's an extraneous solution. Additionally, since the square root is always non-negative, the right side of the equation, x + 3, must also be non-negative. This gives us another constraint: x + 3 ≥ 0, which means x ≥ -3. Combining these constraints, we know our solution, if it exists, must lie in the interval -3 ≤ x ≤ 1/3. Understanding these limitations from the start will help us avoid pitfalls and ensure we arrive at the correct answer. It's like setting the boundaries of our search area before we even begin the treasure hunt!
Isolating the Square Root
The first crucial step in solving any equation involving radicals is to isolate the radical term. In our case, the equation √1-3x = x + 3 already has the square root term isolated on the left side. This is excellent news because it means we can immediately proceed to the next step, which is eliminating the square root. Isolating the square root is essential because it allows us to apply the inverse operation (squaring, in this case) to both sides of the equation, effectively removing the radical. If we didn't isolate the square root first, we'd end up with a much more complicated expression when we square both sides, making the equation significantly harder to solve. Think of it like preparing the foundation before building a house; isolating the square root sets the stage for a smoother solution process. This seemingly simple step is a cornerstone of solving radical equations, and mastering it will make the rest of the process much more manageable. So, remember, always isolate the radical first!
Squaring Both Sides
Now that we have the square root isolated, the next step is to eliminate it by squaring both sides of the equation. Squaring both sides of √1-3x = x + 3 gives us (√1-3x)² = (x + 3)². This simplifies to 1-3x = (x + 3)². Remember, squaring both sides is a valid operation as long as we keep in mind that it can sometimes introduce extraneous solutions, which we'll need to check later. The left side of the equation becomes simply 1-3x, as the square root and the square cancel each other out. The right side, however, requires a bit more attention. We need to expand (x + 3)², which means multiplying (x + 3) by itself. This gives us x² + 6x + 9. So, our equation now looks like 1-3x = x² + 6x + 9. We've successfully eliminated the square root, but now we're dealing with a quadratic equation. Don't worry, we're well-equipped to handle this! Squaring both sides is a powerful technique, but it's crucial to remember the potential for extraneous solutions. We're one step closer to solving the puzzle!
Simplifying to a Quadratic Equation
After squaring both sides, we arrived at the equation 1-3x = x² + 6x + 9. The next step is to simplify this into a standard quadratic equation form, which is ax² + bx + c = 0. To do this, we need to move all the terms to one side of the equation, leaving zero on the other side. Let's subtract 1 and add 3x to both sides of the equation. This gives us 0 = x² + 6x + 9 - 1 + 3x. Combining like terms, we get 0 = x² + 9x + 8. Now we have a quadratic equation in the standard form, where a = 1, b = 9, and c = 8. This form is incredibly useful because it allows us to use various methods to solve for x, such as factoring, completing the square, or using the quadratic formula. Simplifying to a quadratic equation is a crucial step because it transforms the problem into a familiar format that we have well-established techniques for solving. It's like converting a complex message into a code we can decipher! With our quadratic equation in hand, we're ready to move on to the next phase: finding the roots.
Solving the Quadratic Equation
Now that we have the quadratic equation x² + 9x + 8 = 0, we need to find the values of x that satisfy this equation. There are several methods we can use, but factoring is often the quickest if the quadratic expression can be factored easily. In this case, we're looking for two numbers that multiply to 8 and add up to 9. Those numbers are 1 and 8. So, we can factor the quadratic equation as (x + 1)(x + 8) = 0. This factorization tells us that the equation will be true if either x + 1 = 0 or x + 8 = 0. Solving these two linear equations, we get x = -1 and x = -8. These are our potential solutions. However, remember that squaring both sides earlier might have introduced extraneous solutions, so we need to check these values in the original equation. Factoring is a powerful tool for solving quadratic equations, and it's a skill worth mastering. It's like finding the hidden key that unlocks the solution! We've found our potential answers, but the journey isn't over yet. We must now verify these solutions.
Checking for Extraneous Solutions
We've arrived at two potential solutions: x = -1 and x = -8. But remember, squaring both sides of an equation can sometimes lead to extraneous solutions, which are values that satisfy the transformed equation but not the original one. This is why it's absolutely crucial to check our solutions in the original equation, √1-3x = x + 3. Let's start with x = -1. Plugging this into the original equation, we get √1-3(-1) = -1 + 3, which simplifies to √4 = 2, and further to 2 = 2. This is a true statement, so x = -1 is indeed a valid solution. Now let's check x = -8. Plugging this into the original equation, we get √1-3(-8) = -8 + 3, which simplifies to √25 = -5, and further to 5 = -5. This is a false statement, so x = -8 is an extraneous solution and must be discarded. Checking for extraneous solutions is a critical step in solving radical equations. It's like double-checking your work to ensure accuracy. By performing this check, we avoid including incorrect answers and ensure that our solution is valid. So, always remember to verify your solutions in the original equation!
The Final Solution
After meticulously solving the equation √1-3x = x + 3 and checking for extraneous solutions, we've arrived at our final answer. We initially found two potential solutions: x = -1 and x = -8. However, upon substituting these values back into the original equation, we discovered that x = -8 is an extraneous solution, meaning it doesn't satisfy the original equation. The only solution that holds true is x = -1. This is the value that makes the equation √1-3x = x + 3 a true statement. Therefore, the solution to the equation is x = -1. It's important to remember that solving equations, especially those involving radicals, is not just about finding potential answers but also about verifying their validity. This careful approach ensures we arrive at the correct and complete solution. So, the final answer is clear: x = -1.
In conclusion, solving the equation √1-3x = x + 3 involves several key steps: isolating the square root, squaring both sides, simplifying to a quadratic equation, solving the quadratic equation, and, most importantly, checking for extraneous solutions. By following these steps carefully, we can confidently arrive at the correct solution, which in this case is x = -1. Remember, mathematics is not just about finding the answer; it's about understanding the process. Keep practicing, and you'll become a master problem-solver!
