Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving a quadratic equation. Quadratic equations might seem intimidating at first, but with a step-by-step approach, they can become much easier to handle. We'll break down the process, making it super clear and understandable for everyone. So, let’s get started and tackle this math problem together! We'll walk through each step, ensuring you grasp the core concepts and can confidently solve similar equations on your own. Remember, the key to mastering math is practice, so let's jump right in!

Understanding the Problem

Before we jump into solving, let's make sure we understand the problem. We have the equation 6(2x + 4)² = (2x + 4) + 2. Our goal is to find the value(s) of x that make this equation true. This looks like a quadratic equation, but it's not in the standard form (ax² + bx + c = 0) yet. So, our first step is to manipulate the equation to get it into that standard form. You might be wondering, why do we need the standard form? Well, it's because the standard form allows us to easily identify the coefficients a, b, and c, which are crucial for solving the equation using various methods like factoring, completing the square, or the quadratic formula. Think of it like organizing your tools before starting a project; having everything in the right place makes the job much easier! Recognizing the structure of the equation is the first key step. This involves understanding that the equation has a squared term, a linear term, and a constant term, all of which need to be brought together on one side to equal zero. This preparation is crucial for applying the correct methods to solve for x. Once we have the equation in standard form, we can move forward with a clear plan.

Step 1: Simplify and Rearrange the Equation

Okay, the first thing we need to do is simplify and rearrange our equation. We've got 6(2x + 4)² = (2x + 4) + 2. Let's start by expanding the squared term. Remember, (2x + 4)² means (2x + 4) * (2x + 4). When we multiply that out, we get 4x² + 16x + 16. Now, we multiply this by 6, giving us 24x² + 96x + 96. On the right side of the original equation, we have (2x + 4) + 2, which simplifies to 2x + 6. So now our equation looks like this: 24x² + 96x + 96 = 2x + 6. To get the equation into the standard quadratic form (ax² + bx + c = 0), we need to move all the terms to one side. Let's subtract 2x and 6 from both sides. This gives us 24x² + 94x + 90 = 0. Now, we have a quadratic equation in the standard form, which is great! But, the numbers are still a bit large. We can simplify this further by dividing the entire equation by their greatest common divisor, which is 2. Dividing each term by 2, we get 12x² + 47x + 45 = 0. This simplified equation is much easier to work with, and it’s mathematically equivalent to the original, just in a more manageable form. This step is vital because it reduces the complexity of the coefficients, making subsequent steps like factoring or using the quadratic formula much smoother.

Step 2: Solving the Quadratic Equation

Now that we have our simplified quadratic equation, 12x² + 47x + 45 = 0, we need to solve for x. There are a couple of ways we can do this: factoring or using the quadratic formula. Factoring is a great method if we can easily find two binomials that multiply to give us our quadratic equation. However, in this case, factoring might be a bit tricky. So, let's use the quadratic formula, which always works! The quadratic formula is: x = [-b ± √(b² - 4ac)] / (2a). In our equation, a = 12, b = 47, and c = 45. Plugging these values into the formula, we get: x = [-47 ± √(47² - 4 * 12 * 45)] / (2 * 12). Let's simplify this. First, calculate the discriminant (the part under the square root): 47² - 4 * 12 * 45 = 2209 - 2160 = 49. So, our equation becomes: x = [-47 ± √49] / 24. The square root of 49 is 7, so we have: x = [-47 ± 7] / 24. Now we have two possible solutions for x. Let's calculate them separately.

Step 3: Calculate the Solutions

Okay, let's calculate our two solutions for x using the quadratic formula we simplified in the last step: x = [-47 ± 7] / 24. First, we'll take the plus sign: x = (-47 + 7) / 24 = -40 / 24. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8. So, x = -5 / 3. Great, we have one solution! Now, let's take the minus sign: x = (-47 - 7) / 24 = -54 / 24. Again, we can simplify this fraction. The greatest common divisor of 54 and 24 is 6. Dividing both by 6, we get x = -9 / 4. So, our two solutions are x = -5/3 and x = -9/4. It's always a good idea to check these solutions by plugging them back into the original equation to make sure they work. This step helps prevent errors and confirms that our calculations are correct. In practical terms, double-checking your work is like proofreading an important document before you send it; it's a final measure to ensure accuracy.

Step 4: State the Answer

Alright, we've done the hard work, and now we have our solutions! We found that x = -5/3 and x = -9/4. So, to answer the original question, the values of x that satisfy the equation 6(2x + 4)² = (2x + 4) + 2 are x = -5/3 or x = -9/4. These are the roots of the quadratic equation, meaning they are the points where the parabola intersects the x-axis if you were to graph the equation. It's awesome when we can break down a problem step by step and arrive at a clear, correct answer! This process demonstrates the power of methodical problem-solving in mathematics. Remember, every complex problem can be simplified into manageable steps, and with practice, you can tackle even the trickiest equations with confidence. The key is to understand the underlying principles and apply them systematically. So, next time you encounter a quadratic equation, remember this guide, and you'll be well-equipped to solve it!

Conclusion

So, guys, we've successfully solved the quadratic equation 6(2x + 4)² = (2x + 4) + 2! We simplified the equation, used the quadratic formula, and found the solutions x = -5/3 and x = -9/4. Remember, the key to solving quadratic equations is to break them down into smaller, manageable steps. First, get the equation into standard form. Then, decide whether to factor or use the quadratic formula. Finally, carefully calculate your solutions and double-check your work. Quadratic equations might seem tough at first, but with practice and a step-by-step approach, you can conquer them. Keep practicing, and you'll become a quadratic equation-solving pro in no time! If you ever get stuck, remember the resources and methods we discussed, and don’t hesitate to seek help or review the steps again. Math is like building a tower—each concept builds on the previous one, so mastering the basics is crucial for tackling more complex problems. Keep exploring, keep learning, and most importantly, keep practicing!