Solving For X: √(81 - 10x) = X - 5 - Step-by-Step Guide
Hey there, math enthusiasts! Ever stumbled upon an equation that looks like it's straight out of a mathematical maze? Well, today we're tackling one of those: √(81 - 10x) = x - 5. Don't worry, we're going to break it down step by step, so you'll be solving these like a pro in no time. Whether you're prepping for an exam, brushing up on your algebra skills, or just love a good math challenge, this guide is for you. So, grab your pencils, and let's dive in!
Understanding the Equation
Before we jump into solving, let's take a moment to understand what we're dealing with. The equation √(81 - 10x) = x - 5 involves a square root, which might seem a bit intimidating at first. But fear not! We'll tame this beast with some basic algebraic techniques. The key here is to isolate the variable x, but we need to get rid of that square root first. Remember, the square root is essentially asking, "What number, when multiplied by itself, equals the expression inside the root?" To undo this, we'll use the inverse operation: squaring.
The Importance of Squaring Both Sides
Now, a crucial rule in algebra is that whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced and ensures we're finding the correct solutions. So, when we square both sides of √(81 - 10x) = x - 5, we're not just making things up; we're applying a legitimate mathematical operation. This is a fundamental principle that underpins much of equation solving, and it's super important to grasp. Think of it like a scale – if you add weight to one side, you need to add the same weight to the other to keep it level. Similarly, in equations, performing the same operation on both sides maintains the equality.
Potential Pitfalls: Extraneous Solutions
However, squaring both sides can sometimes introduce what we call extraneous solutions. These are solutions that satisfy the transformed equation (after squaring) but don't actually work in the original equation. Why? Because squaring can turn negative numbers into positive ones, potentially masking inconsistencies. So, it's absolutely vital that we check our solutions at the end by plugging them back into the original equation. This step is our safety net, ensuring we only accept the true solutions and discard the imposters. Extraneous solutions are a common trap in radical equations, so always be vigilant!
Step-by-Step Solution
Okay, let's get our hands dirty and walk through the solution step by step. We'll break it down into manageable chunks, so you can follow along easily. Remember, the key is to understand the logic behind each step, not just memorize the process. Once you grasp the 'why', the 'how' becomes much clearer.
Step 1: Squaring Both Sides
As we discussed, our first move is to eliminate the square root. To do this, we square both sides of the equation √(81 - 10x) = x - 5. This gives us:
(√(81 - 10x))² = (x - 5)²
The square root and the square cancel each other out on the left side, leaving us with:
81 - 10x = (x - 5)²
Now, we need to expand the right side. Remember the formula for squaring a binomial: (a - b)² = a² - 2ab + b². Applying this to (x - 5)², we get:
81 - 10x = x² - 10x + 25
Step 2: Rearranging the Equation
Our goal is to solve for x, and to do that, we need to rearrange the equation into a standard quadratic form, which is ax² + bx + c = 0. This makes it easier to apply methods like factoring or the quadratic formula. To achieve this, we'll move all the terms to one side of the equation. Let's subtract 81 and add 10x to both sides:
81 - 10x - 81 + 10x = x² - 10x + 25 - 81 + 10x
This simplifies to:
0 = x² - 56
Step 3: Solving the Quadratic Equation
Now we have a quadratic equation in a simplified form: x² - 56 = 0. There are several ways to solve quadratic equations, but in this case, the easiest method is to isolate x² and then take the square root of both sides. Let's add 56 to both sides:
x² = 56
Now, we take the square root of both sides. Remember that when we take the square root, we need to consider both the positive and negative roots:
x = ±√56
We can simplify √56 by factoring out the perfect square 4: √56 = √(4 * 14) = 2√14. So, our solutions are:
x = 2√14 and x = -2√14
Step 4: Checking for Extraneous Solutions
This is the most crucial step! We need to plug our potential solutions back into the original equation, √(81 - 10x) = x - 5, to see if they actually work. Let's start with x = 2√14:
√(81 - 10(2√14)) = 2√14 - 5
This looks a bit messy, but let's approximate the values. √14 is approximately 3.74, so 2√14 is about 7.48. Plugging this in:
√(81 - 10(7.48)) ≈ 7.48 - 5
√(81 - 74.8) ≈ 2.48
√6.2 ≈ 2.48
2.49 ≈ 2.48
This solution seems to hold up. Now let's check x = -2√14, which is approximately -7.48:
√(81 - 10(-2√14)) = -2√14 - 5
√(81 + 10(7.48)) ≈ -7.48 - 5
√(81 + 74.8) ≈ -12.48
√155.8 ≈ -12.48
Here, we run into a problem. The square root of a positive number cannot be negative. So, x = -2√14 is an extraneous solution.
The Final Answer
After all that work, we've arrived at our final answer. The only valid solution to the equation √(81 - 10x) = x - 5 is:
x = 2√14
Or, approximately:
x ≈ 7.48
Key Takeaways and Common Mistakes
Alright, guys, let's recap what we've learned and highlight some common pitfalls to avoid. Solving equations with square roots can be tricky, but with a systematic approach, you can conquer them!
Key Takeaways:
- Squaring Both Sides: This is the fundamental step to eliminate the square root. Remember to square the entire side, not just individual terms.
- Rearranging into Quadratic Form: Get the equation into the form ax² + bx + c = 0 to make it easier to solve.
- Checking for Extraneous Solutions: This is the most critical step. Always plug your solutions back into the original equation.
Common Mistakes:
- Forgetting to Square the Entire Side: When squaring (x - 5), remember it's (x - 5)(x - 5), not x² - 5².
- Skipping the Extraneous Solution Check: This is a recipe for disaster! Extraneous solutions are sneaky, so don't skip this step.
- Incorrectly Applying the Order of Operations: Make sure you're following PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) when simplifying.
Practice Makes Perfect
So, there you have it! We've successfully solved the equation √(81 - 10x) = x - 5. Remember, the key to mastering these types of problems is practice. The more you work through them, the more comfortable you'll become with the process. Try tackling similar equations, and don't be afraid to make mistakes – that's how we learn!
Resources for Further Practice
If you're looking for more practice problems, there are tons of resources available online and in textbooks. Websites like Khan Academy, Mathway, and Purplemath offer a wealth of examples and exercises. You can also find practice problems in algebra textbooks or workbooks. The important thing is to consistently challenge yourself and reinforce your understanding.
Tips for Success
- Break it Down: Complex problems can seem less daunting when you break them into smaller, more manageable steps.
- Show Your Work: Write out each step clearly. This helps you keep track of your progress and makes it easier to spot mistakes.
- Check Your Answers: Always double-check your work, especially when dealing with square roots and quadratic equations.
- Don't Give Up: Math can be challenging, but with perseverance, you can overcome any obstacle.
Conclusion
Well, guys, we've reached the end of our mathematical journey for today. I hope this step-by-step guide has helped you understand how to solve equations with square roots, specifically √(81 - 10x) = x - 5. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep exploring, and most importantly, keep having fun with it! Until next time, happy solving!