Solving X² - 3x - 6 = 0 A Step-by-Step Guide
Hey guys! Today, we're diving into the world of quadratic equations. Don't worry, it's not as scary as it sounds! We're going to break down how to solve the equation x² - 3x - 6 = 0, step by step. So, grab your pencils and let's get started!
Understanding Quadratic Equations
Before we jump into solving, let's make sure we're all on the same page. A quadratic equation is basically an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers), and 'x' is our variable. The highest power of 'x' in a quadratic equation is always 2. In our case, x² - 3x - 6 = 0, we can see that 'a' is 1, 'b' is -3, and 'c' is -6. Recognizing this standard form is crucial because it helps us identify the different parts of the equation and choose the right method for solving it.
Now, why are we even bothering with these quadratic equations? Well, they pop up everywhere in the real world! From calculating the trajectory of a ball thrown in the air to designing bridges and even modeling population growth, quadratic equations are super useful tools. Understanding how to solve them opens up a whole new world of problem-solving possibilities. There are several methods we can use, like factoring, completing the square, or using the quadratic formula. Each method has its strengths and weaknesses, and the best one to use often depends on the specific equation you're dealing with. For example, factoring is great when the equation can be easily broken down into simpler terms, while the quadratic formula is a more general approach that always works, no matter how messy the equation looks. And completing the square? It's a powerful technique that not only helps solve equations but also gives us a deeper understanding of the structure of quadratic expressions. So, buckle up, because we're about to explore the magic of quadratic equations!
Choosing the Right Method
Okay, so we've got our equation, x² - 3x - 6 = 0. The big question is: how do we solve it? We have a few options in our toolbox, but let's think about the best approach for this particular equation. Factoring is often the first method people try because it can be the quickest. However, it only works if we can easily find two numbers that multiply to 'c' (which is -6 in our case) and add up to 'b' (which is -3). Hmm, let's see… the factors of -6 are (1, -6), (-1, 6), (2, -3), and (-2, 3). None of these pairs add up to -3, so factoring isn't going to work for us this time. Don't worry, it's not a dead end! This is just part of the problem-solving process. Sometimes, you try one method, and it doesn't quite fit. That's perfectly okay! It just means we need to reach for a different tool.
Next up, we could try completing the square. This method is a bit more involved, but it's super powerful. It involves manipulating the equation to create a perfect square trinomial, which we can then easily solve. While completing the square always works, it can be a bit tedious, especially when the coefficients (the numbers in front of the 'x' terms) are fractions. In our equation, the coefficient of x² is 1, which is nice, but the coefficient of x is -3, which isn't a fraction, but it's also not an even number. This means that completing the square might involve some fractions along the way, making the process a bit more cumbersome. So, while it's a viable option, there might be an even better choice for us.
That brings us to the quadratic formula! This is our trusty, always-works, no-matter-what method. It's like the Swiss Army knife of quadratic equation solving. The quadratic formula is a general solution that we can use for any quadratic equation in the form ax² + bx + c = 0. It looks a bit intimidating at first, but once you get the hang of it, it's actually quite straightforward. The formula is: x = (-b ± √(b² - 4ac)) / 2a. See? Not so scary! Because it's a reliable and universally applicable method, and because factoring didn't work out so easily, the quadratic formula is going to be our weapon of choice for solving x² - 3x - 6 = 0.
Applying the Quadratic Formula
Alright, let's get down to business and use the quadratic formula to solve our equation, x² - 3x - 6 = 0. Remember, the quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. First things first, we need to identify our 'a', 'b', and 'c' values. As we discussed earlier, in our equation, a = 1, b = -3, and c = -6. Now, it's just a matter of plugging these values into the formula and doing the math. This is where careful attention to detail is key. It's super easy to make a small mistake with the signs or the order of operations, which can throw off your entire answer. So, take your time, double-check your work, and let's do this!
Let's start by substituting the values into the formula: x = (-(-3) ± √((-3)² - 4 * 1 * -6)) / (2 * 1). Notice how we've carefully replaced each variable with its corresponding value, paying close attention to the negative signs. This is a critical step to avoid errors. Now, we need to simplify the expression. First, let's deal with the negative signs: -(-3) becomes 3. So, we have x = (3 ± √((-3)² - 4 * 1 * -6)) / (2 * 1). Next, let's simplify the exponent: (-3)² is 9. Our equation now looks like this: x = (3 ± √(9 - 4 * 1 * -6)) / (2 * 1). Now, let's take care of the multiplication inside the square root: 4 * 1 * -6 is -24. So, we have x = (3 ± √(9 - (-24))) / (2 * 1). Remember that subtracting a negative number is the same as adding a positive number, so 9 - (-24) becomes 9 + 24, which is 33. Our equation is getting simpler: x = (3 ± √33) / (2 * 1). Finally, let's simplify the denominator: 2 * 1 is 2. So, we have x = (3 ± √33) / 2.
The Two Solutions
Okay, we've arrived at a key point! Notice the ± sign in our solution, x = (3 ± √33) / 2. This little symbol is super important because it tells us that we actually have two solutions to our quadratic equation. One solution comes from using the plus sign, and the other comes from using the minus sign. This is a fundamental property of quadratic equations – they often have two distinct solutions (although sometimes those solutions can be the same number). These solutions represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis. So, let's break it down and find those two solutions.
First, let's find the solution using the plus sign: x = (3 + √33) / 2. √33 is an irrational number, meaning it can't be expressed as a simple fraction. Its decimal approximation is about 5.74. So, x ≈ (3 + 5.74) / 2. Adding 3 and 5.74 gives us 8.74, and dividing that by 2 gives us approximately 4.37. So, one solution is approximately x ≈ 4.37. Now, let's find the solution using the minus sign: x = (3 - √33) / 2. Again, we'll use the approximation √33 ≈ 5.74. So, x ≈ (3 - 5.74) / 2. Subtracting 5.74 from 3 gives us -2.74, and dividing that by 2 gives us approximately -1.37. So, our other solution is approximately x ≈ -1.37. And there you have it! We've successfully solved the quadratic equation x² - 3x - 6 = 0 using the quadratic formula, and we've found two solutions: x ≈ 4.37 and x ≈ -1.37. Remember, these are approximate solutions because we used a decimal approximation for √33. If we wanted the exact solutions, we would leave them in the form (3 + √33) / 2 and (3 - √33) / 2.
Checking Our Answers
We've done the hard work of solving the equation, but before we declare victory, it's always a smart idea to check our answers. This is a crucial step in any mathematical problem-solving process, not just for quadratic equations. Checking our solutions helps us catch any potential errors we might have made along the way, and it gives us confidence that our answers are correct. There are a couple of ways we can check our solutions. One way is to plug our solutions back into the original equation and see if they make the equation true. If we substitute our solutions for 'x' in the equation x² - 3x - 6 = 0, the left side of the equation should equal zero. The other way is to use graphing calculator or an online tool like Desmos. We can plot the graph of y=x² - 3x - 6. The solutions should be where the graph crosses x-axis.
Let's start by plugging in our approximate solutions, x ≈ 4.37 and x ≈ -1.37, into the original equation. For x ≈ 4.37, we have: (4.37)² - 3(4.37) - 6 ≈ 0. Calculating this, we get 19.0969 - 13.11 - 6 ≈ -0.0131. This is very close to zero, which is what we expect. The small difference is due to the fact that we're using an approximate value for x. Now, let's try x ≈ -1.37: (-1.37)² - 3(-1.37) - 6 ≈ 0. Calculating this, we get 1.8769 + 4.11 - 6 ≈ -0.0131. Again, this is very close to zero, which confirms that our solutions are likely correct. Another very important reason why checking answers is so important because it trains our brains to think critically and analytically about problems, it reinforces our understanding of the concepts, and it helps us develop a growth mindset. We see mistakes not as failures, but as opportunities to learn and improve. So, let’s celebrate this last step, knowing that we’ve not only solved the problem but also honed our skills as problem-solvers!
Conclusion
Fantastic job, guys! We've successfully navigated the world of quadratic equations and solved x² - 3x - 6 = 0 using the quadratic formula. We discussed why understanding quadratic equations is important, explored different methods for solving them, and learned how to apply the quadratic formula step by step. We even checked our answers to make sure they were correct! Remember, the quadratic formula is a powerful tool that can solve any quadratic equation, so keep it in your mathematical toolbox. And most importantly, don't be afraid to tackle challenging problems – with practice and the right approach, you can conquer anything! Keep practicing, keep exploring, and keep learning!
