Solving X In The Equation X^2 + 2x + 1 = 17 A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of quadratic equations, specifically focusing on how to solve for 'x' in the equation x² + 2x + 1 = 17. Quadratic equations might seem intimidating at first, but with a systematic approach, they can be cracked quite easily. So, let's roll up our sleeves and get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are all about. In essence, a quadratic equation is a polynomial equation of the second degree. This means that 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 zero. Understanding this standard form is crucial because it lays the foundation for various solving techniques. Remember, the goal when solving any equation is to find the value(s) of the variable that make the equation true. For quadratic equations, this often means finding the values of 'x' that satisfy the equation. These values are also known as the roots or solutions of the equation. So, keep this in mind as we move forward – we're on a quest to find those elusive 'x' values!

Key characteristics of quadratic equations: The presence of the x² term, the possibility of having two solutions (roots), and the various methods available to solve them, such as factoring, completing the square, and the quadratic formula, make quadratic equations a fundamental topic in algebra. Recognizing these characteristics helps in identifying and tackling quadratic equations effectively. It's like knowing the traits of a puzzle before trying to solve it – you're better equipped to find the pieces that fit.

Transforming the Equation

Now, let's bring our attention back to the equation at hand: x² + 2x + 1 = 17. The first step in tackling this equation is to transform it into the standard quadratic form, which, as we discussed, is ax² + bx + c = 0. To achieve this, we need to get all the terms on one side of the equation and set it equal to zero. In our case, this means subtracting 17 from both sides of the equation. This seemingly simple step is a game-changer because it aligns our equation with the standard form, making it easier to apply various solving methods. It's like preparing the canvas before you start painting – you're setting the stage for the masterpiece to come!

After subtracting 17, our equation becomes x² + 2x + 1 - 17 = 0, which simplifies to x² + 2x - 16 = 0. Ta-da! We've successfully transformed our equation into the standard quadratic form. Now, we can clearly see that a = 1, b = 2, and c = -16. Identifying these coefficients is a crucial step because it will help us choose the most appropriate method for solving the equation. So, give yourself a pat on the back – you've completed the first critical step in solving for 'x'!

Choosing the Right Method

With our equation now in standard form (x² + 2x - 16 = 0), the next crucial step is to decide which method to use to solve for 'x'. There are several options available, each with its own strengths and weaknesses. The most common methods include factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific characteristics of the equation, such as the values of the coefficients and whether the equation can be easily factored. It's like having a toolbox full of different tools – you need to choose the right one for the job at hand.

Factoring is a great method when the quadratic expression can be easily broken down into two binomial factors. However, not all quadratic equations are factorable, and attempting to factor a non-factorable equation can be a frustrating and time-consuming endeavor. Completing the square is a more versatile method that can be used to solve any quadratic equation. It involves manipulating the equation to create a perfect square trinomial, which can then be easily solved. However, completing the square can be a bit more complex and time-consuming than factoring. The quadratic formula is the most foolproof method, as it can be used to solve any quadratic equation, regardless of whether it is factorable or not. It's a direct formula that gives you the solutions for 'x' by simply plugging in the values of a, b, and c. Given our equation x² + 2x - 16 = 0, factoring doesn't seem straightforward, so we'll explore the other methods.

Why the Quadratic Formula?

Considering our options, the quadratic formula emerges as the most reliable method for solving our equation x² + 2x - 16 = 0. While completing the square is a viable alternative, the quadratic formula offers a direct and efficient route to the solutions, especially when the equation isn't easily factorable. The quadratic formula is like the GPS of quadratic equation solving – it guides you directly to the answer without the need for detours. It's a powerful tool that ensures we can find the solutions for 'x' without getting bogged down in complex manipulations.

The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. This formula might look a bit intimidating at first, but it's actually quite straightforward to use. All we need to do is identify the values of a, b, and c from our equation and plug them into the formula. It's like following a recipe – once you have the ingredients (the values of a, b, and c), you just follow the steps (the formula) to get the final result (the solutions for 'x'). So, let's embrace the quadratic formula and see how it helps us unlock the mystery of 'x' in our equation!

Applying the Quadratic Formula

Now, let's get our hands dirty and apply the quadratic formula to our equation, x² + 2x - 16 = 0. As we've already identified, a = 1, b = 2, and c = -16. The quadratic formula, as a reminder, is: x = (-b ± √(b² - 4ac)) / 2a. The key to successfully using the formula is to carefully substitute the values and follow the order of operations. It's like performing a delicate surgical procedure – precision and attention to detail are paramount.

Let's plug in the values: x = (-2 ± √(2² - 4 * 1 * -16)) / (2 * 1). See how we've replaced a, b, and c with their respective values? Now, let's simplify step by step. First, we calculate the expression inside the square root: 2² - 4 * 1 * -16 = 4 + 64 = 68. So, our equation becomes x = (-2 ± √68) / 2. We're getting closer! The next step is to simplify the square root, if possible. This often involves finding the prime factors of the number under the square root and looking for pairs that can be taken out.

Simplifying the Solution

We've arrived at x = (-2 ± √68) / 2, and now it's time to simplify this expression to its fullest. The key to simplification here lies in understanding how to handle the square root. Remember, simplifying a square root involves finding perfect square factors within the number under the root and extracting them. It's like panning for gold – you're looking for the valuable nuggets (perfect squares) hidden within the ore (the number under the root).

Let's focus on √68. We need to find the largest perfect square that divides 68. The prime factorization of 68 is 2 * 2 * 17, which can be written as 2² * 17. Aha! We have a perfect square factor, 2². This means we can rewrite √68 as √(2² * 17). Now, we can take the square root of 2², which is 2, and bring it outside the square root, leaving us with 2√17. This simplification is a crucial step because it makes our solution more elegant and easier to work with. It's like polishing a rough diamond to reveal its brilliance.

Substituting this back into our equation, we get x = (-2 ± 2√17) / 2. Notice that both terms in the numerator have a common factor of 2. This gives us an opportunity to further simplify the expression by dividing both the -2 and the 2√17 by 2. This final simplification is like putting the finishing touches on a masterpiece – it's the last step that brings everything together and reveals the final, polished solution.

The Final Solutions for x

After our meticulous simplification process, we've arrived at the final solutions for x. Starting from x = (-2 ± 2√17) / 2, we can divide both terms in the numerator by 2, which gives us x = -1 ± √17. And there you have it! We've successfully solved for x in the equation x² + 2x + 1 = 17. These solutions represent the values of x that, when plugged back into the original equation, will make the equation true. It's like finding the key that unlocks a door – these values of x are the keys that unlock the solution to our quadratic equation.

So, what do these solutions actually mean? The ± sign indicates that we have two solutions: one where we add √17 to -1, and another where we subtract √17 from -1. These are the two points where the parabola represented by the equation intersects the x-axis. It's like finding the coordinates on a map – these solutions pinpoint the exact locations where our equation's graph crosses the horizontal axis. Therefore, the two solutions are: x = -1 + √17 and x = -1 - √17. These are the roots of our quadratic equation, the values that satisfy the equation and make it balance perfectly. Congratulations, you've mastered the art of solving for x in this quadratic equation!

Conclusion

In this comprehensive guide, we've journeyed through the process of solving the quadratic equation x² + 2x + 1 = 17. We started by understanding the fundamentals of quadratic equations, transforming our equation into standard form, and then strategically choosing the quadratic formula as our method of attack. We meticulously applied the formula, simplified the resulting expression, and finally arrived at our two solutions: x = -1 + √17 and x = -1 - √17. This journey highlights the power of a systematic approach and the importance of understanding each step in the process. It's like learning a new language – each step builds upon the previous one, leading to fluency and mastery.

Solving quadratic equations is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts. It's not just about finding the right answer; it's about understanding the underlying principles and developing problem-solving skills that can be applied in various contexts. So, keep practicing, keep exploring, and keep challenging yourself with new equations. The world of mathematics is vast and fascinating, and quadratic equations are just the beginning of your exciting journey. And remember, every equation solved is a step forward in your mathematical adventure!

Therefore, the correct answer is B. x = -1 ± √17