Solve √(x-4+5)=2: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into a classic square root equation and breaking down the steps to find its solution. We'll tackle the equation √(x - 4 + 5) = 2 with a clear, easy-to-follow approach. So, grab your pencils, and let's get started!
Understanding Square Root Equations
Before we jump into solving our specific equation, let's quickly recap what square root equations are all about. In essence, they are equations where the variable (in our case, 'x') is trapped inside a square root symbol. The key to solving these equations is to isolate the square root and then eliminate it by squaring both sides of the equation. This process allows us to work with a simpler, algebraic expression and ultimately find the value(s) of 'x' that satisfy the original equation. However, it's super important to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that don't actually work when plugged back into the original equation. Therefore, we always need to check our answers at the end to make sure they're legit!
When you encounter square root equations, the first step is crucial: isolate the square root term. This means getting the square root expression all by itself on one side of the equation. Think of it like freeing the variable from its radical prison! To do this, you might need to perform operations like adding, subtracting, multiplying, or dividing terms on both sides of the equation. Once the square root is isolated, the next step is to eliminate the radical by squaring both sides. Squaring a square root effectively cancels it out, leaving you with the expression inside the radical. This transforms the equation into a more manageable form, usually a linear or quadratic equation, which you can then solve using standard algebraic techniques. For instance, if you have an equation like √(x + 3) = 5, you would square both sides to get x + 3 = 25. Then, you can easily solve for x by subtracting 3 from both sides, giving you x = 22. Remember, always double-check your solutions by substituting them back into the original equation to ensure they are valid. This step is crucial because squaring both sides can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. By verifying your solutions, you can confidently identify and discard any extraneous roots.
Step-by-Step Solution of √(x - 4 + 5) = 2
Now, let's tackle our equation: √(x - 4 + 5) = 2. We'll break it down into manageable steps to make it super clear.
Step 1: Simplify the Expression Inside the Square Root
Our first step is to simplify the expression inside the square root. We have x - 4 + 5. Combining the constants, -4 + 5, gives us 1. So, the expression becomes x + 1. Now, our equation looks like this: √(x + 1) = 2.
Step 2: Isolate the Square Root
In this case, the square root is already isolated on the left side of the equation, which makes our job easier! We have √(x + 1) = 2.
Step 3: Eliminate the Square Root by Squaring Both Sides
To get rid of the square root, we'll square both sides of the equation. Squaring √(x + 1) gives us x + 1. Squaring 2 gives us 4. So, our equation now looks like this: x + 1 = 4.
Step 4: Solve for x
Now we have a simple linear equation. To solve for x, we need to isolate it. We can do this by subtracting 1 from both sides of the equation. x + 1 - 1 = 4 - 1, which simplifies to x = 3.
Step 5: Check the Solution
This is a crucial step! We need to plug our solution, x = 3, back into the original equation to make sure it works. Our original equation was √(x - 4 + 5) = 2. Substituting x = 3, we get √(3 - 4 + 5) = 2. Simplifying inside the square root, we have √(4) = 2. And indeed, √4 is equal to 2. So, our solution x = 3 is valid!
Common Mistakes to Avoid
When solving square root equations, there are a few common pitfalls to watch out for. Avoiding these mistakes will help you get to the correct answer every time.
One frequent error is forgetting to check for extraneous solutions. As we mentioned earlier, squaring both sides of an equation can sometimes introduce solutions that don't actually satisfy the original equation. These are called extraneous solutions. To avoid this mistake, always substitute your solutions back into the original equation and verify that they hold true. For example, if you solve an equation and get two possible solutions, say x = 2 and x = -2, you need to plug both values back into the original equation to see if they work. If one of them doesn't, it's an extraneous solution and should be discarded.
Another common mistake is incorrectly squaring both sides of the equation. When squaring an expression with multiple terms, like (a + b)², you need to use the distributive property (or the FOIL method) to expand it correctly. The correct expansion is a² + 2ab + b², not simply a² + b². For instance, if you have the equation √(x + 1) = x - 1, squaring both sides gives you (√(x + 1))² = (x - 1)². The left side simplifies to x + 1, but the right side needs to be expanded as (x - 1)² = x² - 2x + 1. Failing to expand the right side correctly will lead to an incorrect equation and, consequently, wrong solutions. Additionally, it's essential to isolate the square root term before squaring both sides. If the square root is not isolated, squaring both sides can complicate the equation unnecessarily. For example, if you have an equation like 2√(x - 3) + 1 = 7, you should first subtract 1 from both sides and then divide by 2 to isolate the square root before squaring. Trying to square the equation without isolating the square root will make the process much more difficult and increase the chances of making an error. By being mindful of these common mistakes and taking the time to perform each step carefully, you can improve your accuracy and confidence in solving square root equations.
Let's Summarize
So, to solve the equation √(x - 4 + 5) = 2, we followed these steps:
- Simplified the expression inside the square root.
- Isolated the square root (it was already isolated in this case).
- Squared both sides to eliminate the square root.
- Solved the resulting linear equation for x.
- Checked our solution to make sure it was valid.
We found that the solution to the equation is x = 3. Remember, always check your solutions to avoid extraneous roots!
Practice Makes Perfect
The best way to master solving square root equations is to practice! Try solving similar equations on your own. You can change the numbers or add more terms to make it more challenging. The more you practice, the more confident you'll become in your problem-solving skills.
And that's it for this guide! I hope you found this explanation helpful. If you have any questions or want to try out some more examples, feel free to ask. Happy solving!
