Solving (x+6)^2 = 50: A Step-by-Step Guide

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Hey guys! Let's dive into solving this equation. It looks a bit intimidating at first, but trust me, we'll break it down into easy-to-follow steps. Our main goal here is to find the values of x that make the equation (x+6)2=50(x+6)^2 = 50 true. We’ll explore each step thoroughly, ensuring you understand the logic behind every move. So, grab your thinking caps, and let’s get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the equation represents. The equation (x+6)2=50(x+6)^2 = 50 is a quadratic equation in disguise. It's in a form where we can easily use the square root property to find the solutions. Remember, the square root property states that if a2=ba^2 = b, then a=_+ba = \_+ \sqrt{b} or a=βˆ’ba = -\sqrt{b}. This is super important because it tells us that there are potentially two solutions for x in this equation. We need to account for both the positive and negative square roots. Think of it like finding the sides of a square when you know the area; there are two possible side lengths (positive and negative, though in real-world scenarios, we usually consider only the positive one). So, with that in mind, let’s proceed to the step-by-step solution!

Step-by-Step Solution

1. Take the Square Root of Both Sides

The first step in solving the equation (x+6)2=50(x+6)^2 = 50 is to get rid of the square on the left side. How do we do that? By taking the square root of both sides of the equation! This is a crucial step, so pay close attention. When we take the square root, we must remember to consider both the positive and negative roots. This gives us:

(x+6)2=Β±50\sqrt{(x+6)^2} = \pm\sqrt{50}

This simplifies to:

x+6=Β±50x + 6 = \pm\sqrt{50}

Don't forget that little Β±\pm there! It's super important because it signifies that we have two potential paths to explore, leading to two different solutions for x. We're essentially saying that (x + 6) could be either the positive square root of 50 or the negative square root of 50. Neglecting this would mean missing out on one of the solutions, and we definitely don't want that!

2. Simplify the Square Root

Now, let's simplify the square root of 50. We need to find the largest perfect square that divides evenly into 50. Think of perfect squares like 4, 9, 16, 25, 36, and so on. The largest perfect square that divides 50 is 25. So, we can rewrite 50\sqrt{50} as 25βˆ—2\sqrt{25 * 2}. This is a handy trick for simplifying radicals, guys!

Using the property ab=aβˆ—b\sqrt{ab} = \sqrt{a} * \sqrt{b}, we get:

50=25βˆ—2=52\sqrt{50} = \sqrt{25} * \sqrt{2} = 5\sqrt{2}

So, our equation now looks like this:

x+6=Β±52x + 6 = \pm 5\sqrt{2}

See how much cleaner that looks? Simplifying radicals makes the equation much easier to work with and reduces the chances of making mistakes later on. Plus, it's just good mathematical practice to present our answers in the simplest form possible. We're almost there, guys! Just one more step to isolate x completely.

3. Isolate x

To isolate x, we need to get it all by itself on one side of the equation. Currently, we have 'x + 6' on the left side. So, what do we do? We subtract 6 from both sides of the equation. This is a fundamental algebraic move – whatever we do to one side, we must do to the other to keep the equation balanced. Think of it like a see-saw; if you add or remove weight from one side, you need to do the same on the other to keep it level.

Subtracting 6 from both sides gives us:

x=βˆ’6Β±52x = -6 \pm 5\sqrt{2}

And there you have it! We've successfully isolated x. This equation tells us that there are two possible values for x: one where we add 525\sqrt{2} to -6, and another where we subtract 525\sqrt{2} from -6. These are our solutions! We've conquered the equation and found the values that make it true. High five!

The Solutions

So, the solutions to the equation (x+6)2=50(x+6)^2 = 50 are:

x=βˆ’6+52x = -6 + 5\sqrt{2}

and

x=βˆ’6βˆ’52x = -6 - 5\sqrt{2}

These are the exact values of x that satisfy the equation. If you were to plug either of these values back into the original equation, you would find that both sides are equal. It's always a good idea to check your solutions, especially in exams or when dealing with important calculations. Checking your work gives you the confidence that you've done everything correctly and helps catch any potential errors. In this case, plugging these values back in would confirm that they are indeed the correct solutions. Great job, guys! We've successfully navigated this problem.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided and see which one matches our solutions.

The options were:

A. x=6Β±25x = 6 \pm 2\sqrt{5} B. x=6Β±52x = 6 \pm 5\sqrt{2} C. x=βˆ’6Β±52x = -6 \pm 5\sqrt{2} D. x=βˆ’6Β±25x = -6 \pm 2\sqrt{5}

Comparing our solutions (x=βˆ’6+52x = -6 + 5\sqrt{2} and x=βˆ’6βˆ’52x = -6 - 5\sqrt{2}) with the answer choices, we can clearly see that option C matches perfectly. Option C, x=βˆ’6Β±52x = -6 \pm 5\sqrt{2}, is the correct answer. This is awesome! We've not only solved the equation but also correctly identified the answer from the given options. This feeling of accomplishment is what makes math so rewarding, guys. Keep up the great work!

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. Let's go over them so you can avoid falling into these traps.

  1. Forgetting the Β±\pm Sign: This is a big one! When taking the square root of both sides of an equation, it's crucial to remember to include both the positive and negative square roots. As we discussed earlier, neglecting the Β±\pm sign will lead to missing one of the solutions. Always, always, always remember the Β±\pm!

  2. Incorrectly Simplifying the Square Root: Simplifying radicals can be tricky if you're not careful. Make sure you find the largest perfect square that divides the number under the radical. For example, when simplifying 50\sqrt{50}, don't just think of 4 as a factor; think of 25, which is the largest perfect square factor. This will save you steps and reduce the chances of error.

  3. Algebraic Errors: Simple algebraic mistakes, like incorrectly adding or subtracting numbers, can throw off your entire solution. Double-check your work at each step to ensure accuracy. It's often helpful to rewrite each step clearly so you can easily spot any errors.

  4. Not Distributing Correctly: If the equation involves terms outside the parentheses, make sure you distribute them correctly. For example, if you had something like 2(x+3)2=502(x + 3)^2 = 50, you would need to deal with the 2 after you've taken the square root. Remember the order of operations (PEMDAS/BODMAS)!

By being aware of these common pitfalls, you can significantly improve your accuracy and confidence when solving similar equations. Practice makes perfect, guys! So, keep solving problems, and you'll become a pro in no time.

Practice Problems

To really nail this concept, let's try a few practice problems. Solving similar problems is the best way to reinforce what you've learned and build your problem-solving skills. Grab a pencil and paper, and let's get to it!

  1. Solve (xβˆ’4)2=36(x - 4)^2 = 36
  2. Solve (x+2)2=25(x + 2)^2 = 25
  3. Solve (xβˆ’1)2=12(x - 1)^2 = 12

Try solving these equations using the same steps we discussed earlier. Remember to take the square root of both sides, simplify the radicals if necessary, and isolate x. Don't forget the Β±\pm sign! Working through these problems will help solidify your understanding and make you even more comfortable with this type of equation. And hey, if you get stuck, don't worry! Review the steps we covered, and give it another shot. Persistence is key!

The solutions to these problems are:

  1. x=10x = 10 and x=βˆ’2x = -2
  2. x=3x = 3 and x=βˆ’7x = -7
  3. x=1+23x = 1 + 2\sqrt{3} and x=1βˆ’23x = 1 - 2\sqrt{3}

Check your answers against these solutions. If you got them all right, awesome job! You're on your way to mastering quadratic equations. If you missed a few, that's totally okay too. Just review your work, identify where you went wrong, and learn from your mistakes. Every mistake is a learning opportunity, guys! So, keep practicing, and you'll get there.

Conclusion

Alright, guys! We've successfully tackled the equation (x+6)2=50(x+6)^2 = 50. We broke it down step-by-step, from taking the square root of both sides to isolating x and simplifying the solution. We also identified the correct answer choice and discussed common mistakes to avoid. Plus, we even worked through some practice problems to solidify your understanding. Phew! That's a lot of math, and you guys crushed it!

Remember, the key to mastering math is practice and persistence. Don't be afraid to make mistakes; they're part of the learning process. Keep solving problems, keep asking questions, and keep challenging yourself. You've got this! And remember, math can be fun! So, embrace the challenge, and enjoy the journey. Until next time, happy solving!