Solving (x+2)^2 - 5 = 7 Step By Step Guide

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Hey guys! Today, we're diving into a fun little math problem: solving the equation (x+2)^2 - 5 = 7. Now, don't let those parentheses and exponents scare you away. We're going to break it down step by step, making it super easy to understand. Think of it as a puzzle – we just need to figure out the right moves to find our solution. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a good look at our equation: (x+2)^2 - 5 = 7. What does it really mean? Well, we have an unknown value, x, that we're trying to find. This x is hanging out inside a set of parentheses, being added to 2. Then, the whole thing is squared (that's the little ^2), and we subtract 5 from that result. Finally, the whole shebang equals 7. Our mission, should we choose to accept it (and we do!), is to figure out what value(s) of x will make this equation true. In essence, we're reverse-engineering the equation, undoing each operation until we isolate x and discover its hidden value. It's like peeling back the layers of an onion, but instead of tears, we get solutions!

When dealing with equations like this, it's crucial to remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is our roadmap, guiding us on how to simplify and solve the equation. Failing to follow the order of operations can lead to incorrect results, and we definitely don't want that. So, keep PEMDAS/BODMAS in your back pocket as we move forward.

Step-by-Step Solution

Alright, let's get our hands dirty and solve this equation. Remember, we're aiming to isolate x on one side of the equation. Here's how we'll do it:

Step 1: Isolate the Squared Term

Our first goal is to get the squared term, (x+2)^2, all by itself on one side of the equation. Currently, we have a "- 5" hanging out with it. To get rid of the "- 5", we'll do the opposite operation: we'll add 5 to both sides of the equation. Why both sides? Because in math, it's all about balance. Whatever we do to one side, we must do to the other to keep the equation true. So, here we go:

(x+2)^2 - 5 + 5 = 7 + 5

This simplifies to:

(x+2)^2 = 12

Great! We've successfully isolated the squared term. We're one step closer to uncovering the value of x.

Step 2: Take the Square Root of Both Sides

Now, we have (x+2)^2 = 12. To get rid of the square, we need to do the opposite operation: take the square root. Again, we'll do this to both sides to maintain balance:

√[(x+2)^2] = ±√12

Notice the "±" (plus or minus) on the right side. This is super important! When we take the square root of a number, there are actually two possible solutions: a positive and a negative one. For example, both 3 and -3, when squared, equal 9. So, we need to consider both possibilities.

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

x + 2 = ±√12

We can also simplify √12. Since 12 = 4 * 3, and 4 is a perfect square, we can write:

√12 = √(4 * 3) = √4 * √3 = 2√3

So, our equation becomes:

x + 2 = ±2√3

Step 3: Isolate x

We're almost there! The final step is to isolate x. We have x + 2 = ±2√3. To get x by itself, we simply subtract 2 from both sides:

x + 2 - 2 = -2 ± 2√3

This gives us our solutions:

x = -2 ± 2√3

The Two Solutions

Because of the "±", we actually have two solutions for x:

  • x = -2 + 2√3
  • x = -2 - 2√3

These are the two values of x that will make the original equation, (x+2)^2 - 5 = 7, true. We can leave the solutions in this exact form, or we can use a calculator to get approximate decimal values:

  • x ≈ 1.464
  • x ≈ -5.464

Verification

It's always a good idea to check our answers to make sure we didn't make any mistakes. Let's plug each solution back into the original equation and see if it holds true.

Verification for x = -2 + 2√3

(x+2)^2 - 5 = 7

((-2 + 2√3) + 2)^2 - 5 = 7

(2√3)^2 - 5 = 7

(4 * 3) - 5 = 7

12 - 5 = 7

7 = 7 (This solution checks out!)

Verification for x = -2 - 2√3

(x+2)^2 - 5 = 7

((-2 - 2√3) + 2)^2 - 5 = 7

(-2√3)^2 - 5 = 7

(4 * 3) - 5 = 7

12 - 5 = 7

7 = 7 (This solution checks out too!)

Both solutions work! We've successfully solved the equation.

Common Mistakes to Avoid

Solving equations can be tricky, and there are a few common pitfalls to watch out for. Here are some things to keep in mind:

  • Forgetting the ± when taking the square root: As we discussed earlier, when you take the square root of both sides of an equation, you need to consider both the positive and negative solutions. Forgetting the "±" will lead to missing one of the answers.
  • Incorrectly applying the order of operations: PEMDAS/BODMAS is your friend! Make sure you're performing operations in the correct order. A common mistake is trying to distribute the square before isolating the parentheses, which is a big no-no.
  • Making arithmetic errors: Simple calculation mistakes can throw off your entire solution. Double-check your work, especially when dealing with negative numbers and radicals.
  • Not verifying your solutions: Plugging your answers back into the original equation is the best way to catch mistakes. It only takes a few minutes, and it can save you from getting the wrong answer.

Alternative Methods (If Applicable)

While the method we used above is the most straightforward for this particular equation, there are alternative approaches you could take. For example, you could expand (x+2)^2, which gives you x^2 + 4x + 4, and then rearrange the equation into a quadratic equation in the form ax^2 + bx + c = 0. From there, you could use the quadratic formula to find the solutions.

However, in this case, expanding the equation is unnecessary and makes the problem more complicated. The method we used, isolating the squared term and then taking the square root, is much more efficient. It's always a good idea to choose the simplest and most direct method for solving a problem.

Real-World Applications (If Applicable)

While this specific equation might seem purely mathematical, the underlying concepts are used in many real-world applications. Solving equations is fundamental to fields like physics, engineering, computer science, and economics. For example, you might use similar techniques to calculate the trajectory of a projectile, design a bridge, or model financial markets. The ability to manipulate equations and solve for unknown variables is a powerful tool in many areas of life.

Conclusion

So there you have it! We've successfully solved the equation (x+2)^2 - 5 = 7, found our two solutions, and verified them. We also discussed common mistakes to avoid and alternative methods (although we stuck with the simplest one). Remember, practice makes perfect! The more you solve equations, the more comfortable and confident you'll become. Keep those math muscles flexed, and you'll be tackling even more challenging problems in no time. Great job, guys! You're all equation-solving superstars!