Solving (2.1-0.3x)(7x+4.2)=0: A Step-by-Step Guide
Hey everyone! Let's dive into solving the equation (2.1 - 0.3x)(7x + 4.2) = 0. Don't worry if it looks a bit intimidating at first; we'll break it down into easy-to-understand steps. This is a classic algebra problem involving a product of two expressions equaling zero. The core concept here is the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, to crack this problem, we'll essentially set each factor equal to zero and solve for x. This will give us the values of x that satisfy the original equation. It's all about isolating x and finding its possible values. This kind of equation pops up in various areas of mathematics and even in real-world applications, making it a super important skill to master. Ready to get started? Let's do it!
Step-by-Step Solution
Alright, let's roll up our sleeves and get to work! The equation (2.1 - 0.3x)(7x + 4.2) = 0 has two factors multiplied together, equaling zero. Our goal is to find the values of x that make this true. Following the Zero Product Property, we will set each factor to zero separately. This will give us two simpler equations to solve. Here's how it breaks down, step by step:
Step 1: Set the First Factor to Zero
We'll start with the first factor: 2.1 - 0.3x = 0. Our aim is to isolate x. First, we'll move the constant term, 2.1, to the other side of the equation. Remember, when you move a term across the equals sign, you change its sign. So, the equation becomes: -0.3x = -2.1. Now, to solve for x, we need to get rid of the -0.3 that's multiplying x. We do this by dividing both sides of the equation by -0.3. Doing this gives us: x = -2.1 / -0.3. Let's simplify this calculation. Dividing -2.1 by -0.3 gives us a positive number, and the result is x = 7. So, the first solution to our original equation is x = 7. We've successfully solved for x in the first part of our problem. It's all about carefully manipulating the equation to isolate x. Keep in mind the rules of algebra: what you do to one side, you must do to the other to keep the equation balanced.
Step 2: Set the Second Factor to Zero
Now, let's tackle the second factor: 7x + 4.2 = 0. We follow the same steps as before to isolate x. First, we move the constant term, 4.2, to the other side of the equation by subtracting 4.2 from both sides. This gives us: 7x = -4.2. Next, we need to get rid of the 7 that's multiplying x. We do this by dividing both sides of the equation by 7. This leads us to: x = -4.2 / 7. Now, let's calculate this. Dividing -4.2 by 7 gives us x = -0.6. And there we have it! The second solution to the original equation is x = -0.6. We've now found both values of x that make the original equation true. It's important to note that we've handled both factors separately, applying basic algebraic principles. The Zero Product Property makes this process straightforward, breaking down a complex-looking equation into simpler, manageable steps. With a little practice, this process becomes second nature. The key is to maintain focus on isolating x and keeping track of the signs.
Verifying the Solutions
Okay, awesome work, guys! We've calculated two potential solutions for x: x = 7 and x = -0.6. But, how can we be absolutely sure that these values are correct? That's where verification comes in. It's a super important step to double-check our work. We'll plug each solution back into the original equation (2.1 - 0.3x)(7x + 4.2) = 0 to see if it holds true. If the equation equals zero after plugging in our solution, then we know we've done it right. This process not only confirms our calculations but also helps us understand the equation better. Verification is like the final test, ensuring our answers are accurate and valid. This is a fundamental aspect of problem-solving in algebra. Making sure our solutions work is a key habit to cultivate in any kind of mathematical work. Let’s check each solution one by one.
Verifying x = 7
First, let's substitute x = 7 into the original equation: (2.1 - 0.3 * 7)(7 * 7 + 4.2) = 0. Let's calculate step by step: first, 0.3 * 7 = 2.1, and then 7 * 7 = 49. Substituting these values back into the equation, we get (2.1 - 2.1)(49 + 4.2) = 0. This simplifies to (0)(53.2) = 0, which equals 0 = 0. Because this statement is true, our solution, x = 7, is correct. So, we've successfully verified that x = 7 is a valid solution to the original equation. This confirms that our previous calculations were accurate. Verification ensures we have a clear understanding of how the solution satisfies the equation. It's like a final stamp of approval, which is awesome!
Verifying x = -0.6
Next, let's substitute x = -0.6 into the original equation: (2.1 - 0.3 * -0.6)(7 * -0.6 + 4.2) = 0. Let's break this down. First, 0.3 * -0.6 = -0.18, and then 7 * -0.6 = -4.2. Substituting these values back into the equation, we get (2.1 - (-0.18))(-4.2 + 4.2) = 0. Simplifying further, we have (2.28)(0) = 0, which results in 0 = 0. This statement is true! So, our second solution, x = -0.6, is also correct. With both solutions verified, we are confident in the accuracy of our answers. This step is an important aspect of problem-solving, ensuring that our solutions align with the original problem. Always take the time to verify your solutions; it will make you a better problem-solver.
Conclusion
Alright, we've reached the finish line! We started with the equation (2.1 - 0.3x)(7x + 4.2) = 0, and through some careful steps, we found two solutions for x. By applying the Zero Product Property and isolating x, we discovered that x = 7 and x = -0.6. We also verified our solutions by plugging them back into the original equation to make sure they worked. The process of solving this equation really shows how different parts of algebra come together. We used basic arithmetic, the Zero Product Property, and the skills of isolating and simplifying to get our answer. It is a great example of how mathematical principles can be applied to break down and solve a problem. The techniques we've used here aren't just limited to this specific equation; they are essential tools for solving a wide range of algebraic problems. Remember, practice makes perfect! Keep working through problems, and you'll become more confident and skilled. This example illustrates how a seemingly complicated equation can be broken down into manageable steps, which are very important to improving your problem-solving skills. Congrats on sticking with it to the end. You now have a solid understanding of how to solve these types of equations. Keep up the great work, and keep practicing those skills.