Solving Radical Equations: Find X In √(5x) - √(5x - 16) = 2
Hey guys! Let's dive into solving a radical equation today. Radical equations might seem intimidating at first, but with a step-by-step approach, they become much more manageable. In this article, we're going to break down how to solve the equation √(5x) - √(5x - 16) = 2. We'll go through each step meticulously, ensuring you understand the logic behind it. Remember, the goal is not just to find the answer, but to understand the process. So, grab your pen and paper, and let’s get started!
Understanding Radical Equations
Before we jump into solving our specific equation, let's quickly recap what radical equations are and some key things to keep in mind. Radical equations are equations where the variable is under a radical, most commonly a square root. The main challenge with these equations is dealing with those pesky square roots, which often requires us to isolate them and then square both sides of the equation.
When dealing with square roots, it's crucial to remember that squaring both sides can sometimes introduce extraneous solutions. Extraneous solutions are solutions that you get algebraically, but they don't actually satisfy the original equation. This is why we always, always, always need to check our solutions at the end. Seriously, don't skip this step! It can be the difference between getting the right answer and getting it completely wrong.
Another important thing to remember is that the expression inside the square root must be non-negative (i.e., greater than or equal to zero). This is because the square root of a negative number is not a real number. This restriction will play a role in determining the valid domain for our solutions.
Step-by-Step Solution for √(5x) - √(5x - 16) = 2
Okay, let's get down to business! We’ll break down the solution into manageable steps, so you can follow along easily. Remember, patience and precision are your best friends when solving radical equations.
Step 1: Isolate one of the radicals.
The first step in solving this equation is to isolate one of the square roots. This means getting one of the square root terms by itself on one side of the equation. In our case, it’s easiest to isolate √(5x). We can do this by adding √(5x - 16) to both sides of the equation:
√(5x) - √(5x - 16) + √(5x - 16) = 2 + √(5x - 16)
This simplifies to:
√(5x) = 2 + √(5x - 16)
Step 2: Square both sides of the equation.
Now that we have one radical isolated, we need to get rid of it. The way we do this is by squaring both sides of the equation. Remember, whatever we do to one side, we have to do to the other to maintain the equality:
(√(5x))^2 = (2 + √(5x - 16))^2
On the left side, the square root and the square cancel each other out, leaving us with:
5x
The right side is a bit trickier because we're squaring a binomial (an expression with two terms). We need to remember the formula (a + b)² = a² + 2ab + b². Applying this formula, we get:
(2 + √(5x - 16))^2 = 2^2 + 2 * 2 * √(5x - 16) + (√(5x - 16))^2
Which simplifies to:
4 + 4√(5x - 16) + (5x - 16)
So, our equation now looks like this:
5x = 4 + 4√(5x - 16) + 5x - 16
Step 3: Simplify the equation and isolate the remaining radical.
Next, we need to simplify the equation. We can start by combining like terms on the right side:
5x = 5x + 4√(5x - 16) - 12
Now, let's isolate the remaining radical term. We can do this by subtracting 5x from both sides:
5x - 5x = 5x - 5x + 4√(5x - 16) - 12
This gives us:
0 = 4√(5x - 16) - 12
Next, add 12 to both sides:
12 = 4√(5x - 16)
Finally, divide both sides by 4:
3 = √(5x - 16)
Step 4: Square both sides again.
We still have a radical, so we need to square both sides again to eliminate it:
3^2 = (√(5x - 16))^2
This simplifies to:
9 = 5x - 16
Step 5: Solve for x.
Now we have a simple linear equation. Let's solve for x. Add 16 to both sides:
9 + 16 = 5x - 16 + 16
25 = 5x
Divide both sides by 5:
25 / 5 = 5x / 5
x = 5
Step 6: Check for extraneous solutions.
This is the most crucial step! We need to plug our solution, x = 5, back into the original equation to make sure it works:
√(5x) - √(5x - 16) = 2
√(5 * 5) - √(5 * 5 - 16) = 2
√(25) - √(25 - 16) = 2
5 - √(9) = 2
5 - 3 = 2
2 = 2
Our solution checks out! This means x = 5 is a valid solution.
Final Answer
Therefore, the solution to the equation √(5x) - √(5x - 16) = 2 is:
x = 5
Key Takeaways
Let's recap the key steps we took to solve this radical equation:
- Isolate one of the radicals.
- Square both sides of the equation.
- Simplify and isolate the remaining radical (if necessary).
- Square both sides again (if necessary).
- Solve for x.
- Check for extraneous solutions.
The checking step is absolutely vital because squaring both sides of an equation can introduce solutions that don't actually work in the original equation. Always double-check!
Practice Makes Perfect
The best way to get comfortable with solving radical equations is to practice. Try solving similar problems on your own. You can change the numbers, add more radicals, or even try equations with cube roots. The more you practice, the better you'll become at recognizing patterns and applying the correct steps.
Remember, math is like learning a new language. It takes time, effort, and a willingness to make mistakes and learn from them. Don't get discouraged if you struggle at first. Keep practicing, and you'll get there!
I hope this explanation helped you understand how to solve the radical equation √(5x) - √(5x - 16) = 2. If you have any questions or want to try another example, feel free to ask. Happy solving, guys!