Solving √(x+58) = X+2 A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a puzzle waiting to be solved? Today, we're diving deep into one such equation: √(x+58) = x+2. This isn't just any equation; it's a gateway to understanding how to handle square roots and algebraic manipulations. So, grab your thinking caps, and let's embark on this mathematical journey together!

The Challenge: √(x+58) = x+2

At first glance, this equation might seem a bit intimidating. The presence of a square root adds a layer of complexity, but don't worry, we're going to break it down step by step. Our main goal here is to isolate x, but we can't do that directly with the square root in the way. Remember, the key to solving any equation is to perform the same operations on both sides, keeping everything balanced and fair. In the world of math, fairness is key!

Before we jump into the nitty-gritty, let's talk strategy. The primary obstacle here is the square root. How do we get rid of it? Well, the inverse operation of a square root is squaring. So, our initial move will be to square both sides of the equation. This will eliminate the square root, making the equation look a lot friendlier. But, and this is a big but, squaring both sides can sometimes introduce what we call extraneous solutions. These are solutions that we find through our algebraic manipulations, but they don't actually satisfy the original equation. Think of them as imposters trying to sneak into our solution set. Therefore, it's super crucial that we check our solutions at the end to make sure they're the real deal.

Now, let's get our hands dirty with the actual solving process. We'll start by squaring both sides, then simplify the resulting equation, and finally, we'll solve for x. Remember, patience is a virtue, especially in math. Each step is a piece of the puzzle, and we'll put them together one at a time.

Step-by-Step Solution

Okay, let's get down to business and solve this equation step by step. Remember, the first hurdle is that pesky square root, so we're going to tackle that head-on by squaring both sides of the equation.

1. Squaring Both Sides

Our equation is √(x+58) = x+2. To eliminate the square root, we square both sides:

(√(x+58))^2 = (x+2)^2

On the left side, the square root and the square cancel each other out, leaving us with:

x + 58

On the right side, we need to expand (x+2)^2. Remember, this means (x+2) multiplied by itself. We can use the FOIL method (First, Outer, Inner, Last) or the binomial theorem to expand this. Let's do it:

(x+2)(x+2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

So, our equation now looks like this:

x + 58 = x^2 + 4x + 4

2. Rearranging the Equation

Now that we've squared both sides, we have a quadratic equation. To solve a quadratic equation, we want to set it equal to zero. This means we need to move all the terms to one side. Let's subtract x and 58 from both sides:

0 = x^2 + 4x + 4 - x - 58

Simplifying, we get:

0 = x^2 + 3x - 54

3. Solving the Quadratic Equation

We now have a quadratic equation in the standard form: ax^2 + bx + c = 0, where a = 1, b = 3, and c = -54. There are a few ways to solve quadratic equations: factoring, using the quadratic formula, or completing the square. Factoring is often the quickest method if we can find factors easily. Let's see if we can factor this equation.

We're looking for two numbers that multiply to -54 and add up to 3. After a bit of thought, we can see that 9 and -6 fit the bill (9 * -6 = -54 and 9 + (-6) = 3). So, we can factor the quadratic equation as follows:

0 = (x + 9)(x - 6)

To find the solutions, we set each factor equal to zero:

x + 9 = 0 or x - 6 = 0

Solving for x, we get:

x = -9 or x = 6

4. Checking for Extraneous Solutions

Remember how we talked about extraneous solutions? This is where we need to check our potential solutions in the original equation. We'll plug each value of x back into the original equation, √(x+58) = x+2, and see if it holds true.

Checking x = -9

√( (-9) + 58 ) = (-9) + 2

√(49) = -7

7 = -7

This is not true! So, x = -9 is an extraneous solution.

Checking x = 6

√( 6 + 58 ) = 6 + 2

√(64) = 8

8 = 8

This is true! So, x = 6 is a valid solution.

5. The Final Solution

After all that work, we've found that the only valid solution to the equation √(x+58) = x+2 is x = 6. It's always a good feeling when you nail down the answer, isn't it?

Key Concepts Revisited

Let's take a moment to recap the key concepts we used to solve this equation. This will help solidify your understanding and make you a more confident equation solver!

Squaring Both Sides

We used the principle of squaring both sides to eliminate the square root. This is a common technique when dealing with radical equations. However, it's crucial to remember that squaring both sides can introduce extraneous solutions, so checking your answers is a must.

Quadratic Equations

Our equation transformed into a quadratic equation after squaring. We solved it by factoring, which is a handy method when the quadratic expression can be factored easily. Remember, the goal is to express the quadratic in the form (x + a)(x + b) = 0, then set each factor to zero to find the solutions.

Extraneous Solutions

Extraneous solutions are those sneaky values that pop up during the solving process but don't actually satisfy the original equation. They're a reminder that algebraic manipulations, while powerful, need to be handled with care. Always, always, always check your solutions!

The Importance of Checking

I can't stress this enough: checking your solutions is vital, especially when dealing with equations involving square roots or other radicals. It's the safety net that ensures you don't fall for extraneous solutions. Think of it as the final boss battle in your math game – you've got to defeat it to claim your victory!

Common Pitfalls and How to Avoid Them

Solving equations can be a bit like navigating a maze. There are twists, turns, and potential dead ends. Let's talk about some common pitfalls people encounter when solving equations like this and how you can steer clear of them.

Forgetting to Check for Extraneous Solutions

This is probably the most common mistake. It's easy to get caught up in the solving process and forget to plug your solutions back into the original equation. But trust me, it's worth the extra few minutes. Make checking a habit, and you'll save yourself a lot of headaches.

Incorrectly Expanding Squares

When squaring a binomial like (x+2), it's tempting to just square each term individually (x^2 + 2^2). But remember, (x+2)^2 means (x+2)(x+2), and you need to use FOIL or the distributive property to expand it correctly. A small mistake here can throw off your entire solution.

Sign Errors

Sign errors are like little gremlins that can sabotage your math. Be extra careful when moving terms across the equals sign. Remember, when you move a term, you change its sign. Double-checking your work for sign errors can save you from frustration.

Not Simplifying Properly

Simplifying as you go is a good habit. It keeps your equations manageable and reduces the chance of errors. Combine like terms, reduce fractions, and generally clean things up as you work through the problem. A tidy equation is a happy equation!

Giving Up Too Soon

Some equations take a bit of effort to solve. Don't get discouraged if you don't see the solution immediately. Take a deep breath, review your steps, and try a different approach if necessary. Perseverance is key in math, as in life.

Practice Makes Perfect

The best way to master solving equations is to practice, practice, practice! The more you work through different types of equations, the more comfortable and confident you'll become. Think of it like learning a new language – the more you use it, the more fluent you become.

So, grab some practice problems, dust off your algebra skills, and get solving! And remember, math isn't just about finding the right answer; it's about the journey of problem-solving and the satisfaction of cracking the code. Happy calculating, guys!

Conclusion

In conclusion, we've successfully navigated the equation √(x+58) = x+2, found our solution (x = 6), and learned some valuable lessons along the way. We've reinforced the importance of squaring both sides, solving quadratic equations, checking for extraneous solutions, and avoiding common pitfalls. Remember, math is a journey, not a destination. Embrace the challenges, celebrate the victories, and keep exploring the fascinating world of numbers and equations! You've got this!