Solving Quadratic Equations: Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into the world of quadratic equations. Specifically, we're going to break down how to solve an equation like 2(x+2)² - 4 = 28. It might look a bit intimidating at first, but trust me, with a few simple steps, we can crack this problem and find the correct solutions. So, grab your pencils, and let's get started!
Understanding the Basics: What are Quadratic Equations?
Before we jump into the equation, let's quickly recap what a quadratic equation is. In its simplest form, a quadratic equation is an equation that can be written as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations are characterized by the presence of a variable raised to the power of two (x²). The solutions to these equations are the values of 'x' that make the equation true. Quadratic equations often have two solutions, but in some cases, they can have one or even no real solutions. This is where things get interesting, and the methods for solving them come into play. Recognizing the structure is the first step towards mastering quadratic equations and other complex problems. In this case, our equation 2(x+2)² - 4 = 28 is a quadratic equation that can be simplified into the standard form. Understanding the principles that govern these equations is fundamental for anyone looking to build a strong foundation in mathematics, especially algebra. The importance of mastering this concept cannot be overstated, as it serves as a building block for more complex topics in mathematics. So, let’s begin to unravel the problem given to us!
To solve this particular problem, we'll need to go through a series of algebraic manipulations. It is important to remember that the goal is to isolate 'x' on one side of the equation. This involves several steps, including using the principles of the distribution, combining like terms, and, in some cases, factoring or using the quadratic formula. Each step is crucial, and it’s important to perform the operations correctly to ensure we get accurate solutions. Keeping a clear head and a methodical approach will help us navigate through these algebraic manipulations without getting confused. Remember, practice is key. The more you work through these problems, the more familiar you will become with the process and the easier it will be to solve them. By the time we're done, you'll feel confident in your ability to tackle similar problems.
Let’s get back to the equation, and then solve the question!
Step-by-Step Solution: Unraveling the Equation
Alright, let's take a look at the equation: 2(x + 2)² - 4 = 28. Our mission? To find the values of 'x' that satisfy this equation. We will walk you through, step by step, so that you understand the logic behind each of the steps. We will try to simplify the equation, step by step, which will make it easier to solve. Now, let’s get started!
Step 1: Isolate the Squared Term
Our first goal is to isolate the term containing the square, which is (x + 2)². To do this, we need to get rid of the -4 on the left side. We do this by adding 4 to both sides of the equation. This maintains the balance of the equation, a fundamental rule in algebra. It ensures that the equation remains valid as we move towards our solution.
So, 2(x + 2)² - 4 + 4 = 28 + 4 simplifies to 2(x + 2)² = 32. Great! Now we are one step closer to isolating the squared term. Next, we need to deal with the 2 that is multiplying the squared term.
Step 2: Divide to Simplify
Now, we need to get rid of that 2 that's multiplying the (x + 2)² term. To do this, we divide both sides of the equation by 2. This is the inverse operation of multiplication and is crucial for simplifying the equation. It's like unwinding the layers of an onion – each step gets us closer to the core. Dividing both sides by 2, we get:
2(x + 2)² / 2 = 32 / 2, which simplifies to (x + 2)² = 16. We're making progress. Now we’re looking good to solve for x!
Step 3: Take the Square Root
Next up, we need to get rid of the square on the (x + 2) term. The way to do this is to take the square root of both sides. This is a critical step, as it introduces two possible solutions. Remember that the square root of a number can be both positive and negative. It's important not to forget this when dealing with quadratic equations. Taking the square root of both sides of (x + 2)² = 16, we get:
√((x + 2)²) = ±√16, which simplifies to x + 2 = ±4. See, we’re almost there!
Step 4: Solve for x
We now have two separate equations to solve:
- x + 2 = 4
- x + 2 = -4
Let's solve the first equation. Subtracting 2 from both sides, we get x = 4 - 2, which gives us x = 2. Now, let's solve the second equation. Subtracting 2 from both sides, we get x = -4 - 2, which gives us x = -6.
So, the solutions to the equation 2(x + 2)² - 4 = 28 are x = 2 and x = -6. Congratulations, we've solved the equation! We’ve successfully found the values of 'x' that satisfy the original equation. Let’s move to the next part, to check the answer.
Checking Your Answers: Verification
It is always a good idea to check your answers! To make sure that we are correct, let's plug these values back into the original equation to verify that they work. This is an important step to confirm that our solutions are correct. It’s always satisfying to see that our solution works. Let’s check!
Checking x = 2
Substitute x = 2 into the original equation:
2((2 + 2)²) - 4 = 2(4²) - 4 = 2(16) - 4 = 32 - 4 = 28. This checks out. Our value for x = 2 is correct!
Checking x = -6
Substitute x = -6 into the original equation:
2((-6 + 2)²) - 4 = 2(-4)² - 4 = 2(16) - 4 = 32 - 4 = 28. Another check. Our value for x = -6 is also correct!
Both of our solutions, x = 2 and x = -6, satisfy the original equation. Therefore, we can confidently say that these are indeed the correct solutions. Isn’t it amazing to see how everything fits together? We have successfully verified our solutions, which demonstrates our understanding of the concepts. This step is a testament to the fact that we can trust our solutions. With this, we have now reached the end of the solution!
Conclusion: Wrapping Up
So, there you have it! The solutions to the equation 2(x + 2)² - 4 = 28 are x = 2 and x = -6. We methodically worked through each step, ensuring we understood the logic behind it. This process not only solves the equation but also reinforces the fundamental concepts of algebra. Quadratic equations may initially seem complex, but with a structured approach and a good understanding of the basics, they become manageable. Remember, practice is key. The more you solve these types of equations, the more comfortable and confident you will become. Keep practicing, and you’ll master them in no time!
This is just one example of a quadratic equation, and there are many more types and methods to learn. Keep exploring, keep practicing, and never stop learning. Each equation you solve is a step forward in your mathematical journey. So, keep up the fantastic work and happy solving!
For more practice and a deeper understanding of quadratic equations, check out Khan Academy's algebra resources.
