Solving X² - 5x - 14 = 0 A Step-by-Step Guide With Verification

Hey guys! Let's dive into solving a quadratic equation today. We've got x² - 5x - 14 = 0, and we're going to break it down step by step so you can not only find the solutions but also verify them. This is super important because, in math, knowing you've got the right answer is just as crucial as finding the answer itself. So, let's get started!

Understanding Quadratic Equations

Before we jump into solving, it's good to understand what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. In our equation, x² - 5x - 14 = 0, we can identify a = 1, b = -5, and c = -14. Understanding this standard form helps us apply the correct methods to solve the equation. Quadratic equations pop up all over the place in real-world applications, from physics (think projectile motion) to engineering (designing structures) and even in computer graphics. So, mastering how to solve them is a seriously useful skill.

Why do we care about solving these equations? Well, the solutions (also called roots or zeros) tell us where the parabola represented by the equation intersects the x-axis. This is incredibly useful information in many contexts. For instance, if you're modeling the path of a ball thrown in the air, the roots would tell you where the ball hits the ground. The process of solving quadratic equations involves finding the values of 'x' that make the equation true. There are several methods to do this, and we’ll be focusing on factoring in this guide because it’s a straightforward and insightful approach for equations that factor nicely.

Method 1: Factoring the Quadratic Equation

Okay, let's get to the fun part: solving the equation! For x² - 5x - 14 = 0, we're going to use the factoring method. Factoring involves breaking down the quadratic expression into two binomial expressions. Basically, we want to rewrite the equation in the form (x + p)(x + q) = 0, where 'p' and 'q' are constants. The key here is to find two numbers that multiply to 'c' (-14 in our case) and add up to 'b' (-5 in our case). This might sound a little tricky at first, but with some practice, you'll get the hang of it.

Think of it like a puzzle: we need to find the right pieces that fit together. Let's list the factors of -14: (-1, 14), (1, -14), (-2, 7), and (2, -7). Now, which of these pairs adds up to -5? Bingo! It’s 2 and -7 because 2 + (-7) = -5. So, we can rewrite our equation as (x + 2)(x - 7) = 0. Now we've factored the quadratic expression, which is a huge step forward! The beauty of factoring is that it turns a slightly complicated problem into a simple one. We’ve essentially broken down the quadratic into two linear factors, and we know that if the product of these factors is zero, then at least one of them must be zero. This is the Zero Product Property, and it’s the key to unlocking our solutions.

To recap, factoring is all about finding the right combination of numbers that multiply to the constant term and add up to the coefficient of the linear term. It might take a little trial and error, but once you find the correct factors, solving the equation becomes a breeze. This method is particularly useful when the quadratic expression can be factored easily, as it provides a direct path to the solutions. So, keep practicing your factoring skills – they’ll come in handy!

Finding the Solutions

Now that we've factored our equation into (x + 2)(x - 7) = 0, we're in the home stretch. Remember that Zero Product Property we talked about? It states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for 'x'. First, let's take x + 2 = 0. To solve for 'x', we subtract 2 from both sides of the equation, giving us x = -2. That’s our first solution! Now, let's move on to the second factor, x - 7 = 0. To solve for 'x' in this case, we add 7 to both sides, which gives us x = 7. And there you have it – our second solution!

So, the solutions to the quadratic equation x² - 5x - 14 = 0 are x = -2 and x = 7. These are the values of 'x' that make the equation true. Think of them as the points where the parabola represented by the equation crosses the x-axis. It’s like finding the specific spots where the graph intersects the zero line. These solutions are also often referred to as the roots or zeros of the quadratic equation. Understanding this terminology is important because you’ll encounter it frequently in math and related fields. We've used the Zero Product Property to essentially split our factored equation into two simpler linear equations, which we then solved individually. This is a powerful technique that makes finding the solutions much more manageable.

Now, before we celebrate our success, it's crucial to verify our solutions. In math, it's not enough to just find an answer; you need to make sure it’s the correct answer. This verification step is what sets strong problem-solvers apart and ensures that you're not just going through the motions but actually understanding the process. So, let's roll up our sleeves and get ready to check our work!

Verifying the Solutions

Alright, guys, let's make sure we nailed it! Verifying our solutions is super important. It's like double-checking your work on a test – you want to be confident you've got the right answers. To verify our solutions, we'll plug each value of 'x' we found back into the original equation, x² - 5x - 14 = 0, and see if it holds true. First, let's check x = -2. We substitute -2 for 'x' in the equation: (-2)² - 5(-2) - 14 = 0. This simplifies to 4 + 10 - 14 = 0, which is 14 - 14 = 0. And guess what? It checks out! So, x = -2 is definitely a solution. High five!

Now, let's verify the second solution, x = 7. We substitute 7 for 'x' in the equation: (7)² - 5(7) - 14 = 0. This simplifies to 49 - 35 - 14 = 0, which is 14 - 14 = 0. Awesome! This one checks out too. So, x = 7 is also a valid solution. By plugging our solutions back into the original equation, we've confirmed that they make the equation true. This process is a fundamental part of problem-solving in mathematics and helps build confidence in your results. It’s not just about getting an answer; it’s about knowing you’ve got the right answer.

Why is verification so important? Well, it catches any potential errors you might have made along the way. Maybe you made a small mistake in your factoring, or perhaps you had a sign error in your calculations. Verification helps you identify and correct these mistakes before moving on. It also reinforces your understanding of the problem and the solution process. By going through the steps of verification, you’re solidifying your knowledge and ensuring that you can apply these concepts in the future. So, always remember to verify your solutions – it’s a crucial part of becoming a successful problem-solver!

Conclusion

So, there you have it! We've successfully solved the quadratic equation x² - 5x - 14 = 0 by factoring, and we found the solutions x = -2 and x = 7. We also verified these solutions by plugging them back into the original equation, ensuring that our answers are correct. Solving quadratic equations is a fundamental skill in algebra, and factoring is just one of the many tools you can use to tackle these problems. Whether you're dealing with math problems in school or applying these concepts in real-world situations, understanding how to solve quadratic equations is incredibly valuable.

Remember, the key to mastering any math skill is practice. The more you work through problems, the more comfortable and confident you'll become. Try solving other quadratic equations by factoring, and don't be afraid to explore other methods, such as the quadratic formula or completing the square. Each method has its strengths and weaknesses, and knowing when to use each one is a sign of true mastery.

Most importantly, remember to verify your solutions! It's the final step that ensures you've got the correct answer and solidifies your understanding of the problem. Math is not just about finding the answer; it's about understanding the process and being able to explain why your answer is correct. Keep practicing, keep verifying, and you'll be solving quadratic equations like a pro in no time! And hey, if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available, from teachers and tutors to online tutorials and forums. Happy solving, guys!