Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of quadratic equations, specifically tackling the equation x² - 2x - 35 = 0. Quadratic equations might seem intimidating at first, but with a little guidance, you'll be solving them like a pro. So, let's break it down step by step. Understanding how to solve quadratic equations is a fundamental skill in algebra and is super useful in many areas of math and science. This guide will not only show you the solution but also help you understand the underlying concepts, so you can confidently solve similar problems in the future.

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are all about. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we're trying to find. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These are the values of x that make the equation true. Quadratic equations pop up everywhere, from physics problems involving projectile motion to engineering designs and even financial models. Recognizing and solving them is a key skill in many different fields, so it's really worth getting comfortable with the methods involved. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and disadvantages, depending on the specific equation you're trying to solve. For some equations, factoring might be the quickest and easiest approach, while for others, the quadratic formula might be necessary. Understanding these different methods and knowing when to apply them is crucial for mastering quadratic equations. Remember, the goal is always to find the values of x that satisfy the equation, and by understanding the different techniques available, you'll be well-equipped to tackle any quadratic equation that comes your way. Keep practicing, and you'll soon find that solving quadratic equations becomes second nature!

Method 1: Factoring the Quadratic Equation

One of the most common and often the easiest methods for solving quadratic equations is factoring. Factoring involves breaking down the quadratic expression into two binomials that, when multiplied together, give you the original quadratic expression. This method is particularly useful when the quadratic equation can be easily factored, meaning that the coefficients are relatively small and the factors are integers. In our case, we have the equation x² - 2x - 35 = 0. We need to find two numbers that multiply to -35 (the constant term) and add up to -2 (the coefficient of the x term). After a little thought, you might realize that the numbers 5 and -7 fit the bill perfectly because 5 * (-7) = -35 and 5 + (-7) = -2. Now, we can rewrite the quadratic equation in factored form: (x + 5)(x - 7) = 0. This factored form tells us that either (x + 5) must equal zero or (x - 7) must equal zero, because 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: x + 5 = 0 gives us x = -5, and x - 7 = 0 gives us x = 7. Therefore, the solutions to the quadratic equation x² - 2x - 35 = 0 are x = -5 and x = 7. Factoring is a powerful tool, and with practice, you'll become adept at recognizing when it's the most efficient method to use. Remember to always double-check your factors to ensure they correctly multiply back to the original quadratic expression. It's also important to note that not all quadratic equations can be easily factored, in which case you might need to resort to other methods like completing the square or using the quadratic formula. But when factoring works, it's often the quickest and simplest way to find the solutions!

Method 2: Using the Quadratic Formula

When factoring doesn't seem to work or is too complicated, the quadratic formula is your best friend. The quadratic formula is a universal method that can solve any quadratic equation, regardless of whether it can be factored or not. It's a bit more involved than factoring, but it's a reliable tool to have in your arsenal. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the general form of the quadratic equation ax² + bx + c = 0. In our equation, x² - 2x - 35 = 0, we can identify a = 1, b = -2, and c = -35. Now, let's plug these values into the quadratic formula: x = (-(-2) ± √((-2)² - 4 * 1 * (-35))) / (2 * 1). Simplifying this, we get: x = (2 ± √(4 + 140)) / 2, which further simplifies to: x = (2 ± √144) / 2. Since the square root of 144 is 12, we have: x = (2 ± 12) / 2. Now, we have two possible solutions for x: x = (2 + 12) / 2 = 14 / 2 = 7, and x = (2 - 12) / 2 = -10 / 2 = -5. So, the solutions to the quadratic equation x² - 2x - 35 = 0 are x = 7 and x = -5, which matches the solutions we found using the factoring method. The quadratic formula is a powerful tool, and it's especially useful when dealing with quadratic equations that have irrational or complex roots. It's also a great way to double-check your solutions when you use other methods like factoring. Remember to carefully substitute the values of a, b, and c into the formula, and be mindful of the order of operations when simplifying the expression. With practice, you'll become comfortable using the quadratic formula and appreciate its versatility in solving quadratic equations.

Verification of the Solutions

To ensure that our solutions are correct, it's always a good idea to verify them by plugging them back into the original equation. This step is crucial, especially when you're dealing with more complex equations, as it helps you catch any potential errors you might have made along the way. Let's start with x = -5. Substituting this value into the equation x² - 2x - 35 = 0, we get: (-5)² - 2(-5) - 35 = 25 + 10 - 35 = 35 - 35 = 0*. So, x = -5 is indeed a solution. Now, let's check x = 7. Substituting this value into the equation, we get: (7)² - 2(7) - 35 = 49 - 14 - 35 = 49 - 49 = 0*. Therefore, x = 7 is also a solution. Since both values satisfy the original equation, we can confidently say that our solutions x = -5 and x = 7 are correct. Verification is a simple but important step that can save you from making mistakes and help you build confidence in your problem-solving skills. It's a good habit to get into, especially when you're learning new concepts or dealing with challenging problems. By taking the time to verify your solutions, you're not only ensuring that you have the right answer, but you're also reinforcing your understanding of the underlying principles. So, always remember to double-check your work and verify your solutions whenever possible!

Conclusion

So there you have it, guys! We've successfully solved the quadratic equation x² - 2x - 35 = 0 using two different methods: factoring and the quadratic formula. We found that the solutions are x = -5 and x = 7. Remember, understanding quadratic equations and knowing how to solve them is a valuable skill that will come in handy in many areas of math and science. Keep practicing, and don't be afraid to try different methods until you find the one that works best for you. Whether you prefer factoring or the quadratic formula, the key is to understand the underlying concepts and be comfortable applying them. And always remember to verify your solutions to ensure that you're on the right track. With a little effort and perseverance, you'll be solving quadratic equations like a pro in no time! Keep up the great work, and happy solving!