Solving X² + 2x + 1 = 17 A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of solving a quadratic equation, specifically the equation x² + 2x + 1 = 17. Quadratic equations, characterized by the highest power of the variable being 2, are a fundamental concept in algebra and have numerous applications in various fields, including physics, engineering, and economics. Understanding how to solve them is crucial for anyone pursuing studies or careers in these areas. Our goal is to find the values of x that satisfy the given equation. We will explore different methods to solve this equation, discuss the underlying principles, and provide a step-by-step solution. The equation x² + 2x + 1 = 17 might look intimidating at first, but with the right approach and understanding of algebraic techniques, it can be solved effectively. We'll break down the problem into manageable steps, ensuring a clear and comprehensive understanding of the solution process. This exploration will not only help in solving this particular equation but also provide a framework for tackling other quadratic equations. Let's embark on this mathematical journey and unravel the solution to this intriguing equation.

Understanding Quadratic Equations

Before we dive into solving the equation x² + 2x + 1 = 17, it's essential to understand what quadratic equations are and their general form. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 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. The coefficients a, b, and c play a significant role in determining the nature and solutions of the equation. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. Recognizing this general form is the first step in identifying and solving quadratic equations. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic equations. The solutions to a quadratic equation are also known as roots or zeros, which represent the values of x that make the equation true. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant (b² - 4ac). Understanding the discriminant helps in predicting the nature of the solutions even before solving the equation. In our specific equation, x² + 2x + 1 = 17, we need to manipulate it into the standard form to apply these solution methods effectively. By subtracting 17 from both sides, we can rewrite the equation as x² + 2x - 16 = 0, which now matches the general form and sets the stage for further analysis and solution.

Methods to Solve Quadratic Equations

Several methods can be employed to solve quadratic equations. Each method has its advantages and is suitable for different types of equations. Let's explore some common methods:

  1. Factoring: Factoring involves expressing the quadratic equation as a product of two linear factors. This method is effective when the quadratic expression can be easily factored. For example, the equation x² + 5x + 6 = 0 can be factored into (x + 2)(x + 3) = 0, leading to the solutions x = -2 and x = -3. However, factoring is not always straightforward, especially when the coefficients are large or the roots are not integers.
  2. Completing the Square: Completing the square is a method that transforms the quadratic equation into a perfect square trinomial. This method is particularly useful when the equation is not easily factorable. The general idea is to manipulate the equation into the form (x + p)² = q, where p and q are constants. From this form, the solutions can be easily obtained by taking the square root of both sides. Completing the square can be applied to any quadratic equation, making it a versatile technique.
  3. Quadratic Formula: The quadratic formula is a universal method for solving quadratic equations. It provides the solutions directly from the coefficients a, b, and c in the general form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). The quadratic formula can be used to solve any quadratic equation, regardless of whether it is factorable or not. The discriminant, b² - 4ac, within the formula, determines the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots.

For the equation x² + 2x + 1 = 17, we will primarily focus on using completing the square and the quadratic formula to find the solutions. These methods offer a systematic approach and ensure accurate results.

Solving x² + 2x + 1 = 17 by Completing the Square

Let's apply the completing the square method to solve the equation x² + 2x + 1 = 17. This method involves transforming the equation into a form where we have a perfect square trinomial on one side. Here are the steps:

  1. Rewrite the equation: First, subtract 17 from both sides to get the equation in the standard form: x² + 2x + 1 - 17 = 0, which simplifies to x² + 2x - 16 = 0.
  2. Move the constant term: Move the constant term to the right side of the equation: x² + 2x = 16.
  3. Complete the square: To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. The value we need to add is (b/2)², where b is the coefficient of the x term. In our equation, b = 2, so we need to add (2/2)² = 1² = 1 to both sides: x² + 2x + 1 = 16 + 1.
  4. Factor the perfect square trinomial: The left side is now a perfect square trinomial, which can be factored as (x + 1)²: (x + 1)² = 17.
  5. Take the square root: Take the square root of both sides of the equation: √(x + 1)² = ±√17, which gives us x + 1 = ±√17.
  6. Solve for x: Finally, subtract 1 from both sides to solve for x: x = -1 ± √17.

Thus, the solutions for the equation x² + 2x + 1 = 17 using the completing the square method are x = -1 + √17 and x = -1 - √17. This method provides a clear and step-by-step approach to finding the solutions, and it's particularly useful when the equation is not easily factorable. The process of completing the square involves manipulating the equation to create a perfect square trinomial, which simplifies the equation and allows us to solve for x by taking the square root. This technique is a fundamental tool in algebra and is applicable to a wide range of quadratic equations.

Solving x² + 2x + 1 = 17 Using the Quadratic Formula

Now, let's solve the same equation, x² + 2x + 1 = 17, using the quadratic formula. The quadratic formula is a powerful tool that can solve any quadratic equation in the form ax² + bx + c = 0. Here's how we apply it:

  1. Rewrite the equation: First, rewrite the equation in the standard form by subtracting 17 from both sides: x² + 2x + 1 - 17 = 0, which simplifies to x² + 2x - 16 = 0.
  2. Identify the coefficients: Identify the coefficients a, b, and c in the equation. In our case, a = 1, b = 2, and c = -16.
  3. Apply the quadratic formula: The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). Substitute the values of a, b, and c into the formula: x = (-2 ± √(2² - 4(1)(-16))) / (2(1)).
  4. Simplify the expression: Simplify the expression step by step: x = (-2 ± √(4 + 64)) / 2, which becomes x = (-2 ± √68) / 2.
  5. Simplify the square root: Simplify the square root term. Since 68 = 4 * 17, we have √68 = √(4 * 17) = 2√17. So, x = (-2 ± 2√17) / 2.
  6. Further simplification: Divide both terms in the numerator by 2: x = -1 ± √17.

Thus, the solutions for the equation x² + 2x + 1 = 17 using the quadratic formula are x = -1 + √17 and x = -1 - √17. This method provides a direct and systematic way to find the solutions, regardless of the complexity of the equation. The quadratic formula is a cornerstone of algebra and is essential for solving any quadratic equation. By substituting the coefficients into the formula and simplifying, we can efficiently determine the roots of the equation. In this case, the quadratic formula confirms the solutions we found using the completing the square method, reinforcing the accuracy and reliability of both techniques.

Comparing the Solutions and Choosing the Correct Answer

After solving the equation x² + 2x + 1 = 17 using both completing the square and the quadratic formula, we arrived at the solutions x = -1 + √17 and x = -1 - √17. Now, let's compare these solutions with the given options to choose the correct answer. The options provided are:

A. x = -1 ± √15 B. x = -1 ± √17 C. x = -2 ± 2√5 D. x = -1 ± √13

Comparing our solutions with the given options, we can see that option B, x = -1 ± √17, matches our solutions exactly. The plus-minus sign (±) indicates that there are two solutions: one with addition and one with subtraction. This notation is a concise way of representing both solutions in a single expression. The other options have different values under the square root or different constant terms, making them incorrect. Therefore, the correct answer is B, which aligns perfectly with the solutions we derived using both the completing the square method and the quadratic formula. This comparison step is crucial in ensuring that the solutions we obtained are accurate and match the expected format. It also highlights the importance of understanding the notation and conventions used in expressing mathematical solutions. In this case, the ± notation simplifies the representation of two distinct solutions, making it easier to compare with the given options. By carefully examining the solutions and the options, we can confidently identify the correct answer and validate our problem-solving process.

Conclusion

In conclusion, we have successfully solved the quadratic equation x² + 2x + 1 = 17 using two different methods: completing the square and the quadratic formula. Both methods led us to the same solutions, x = -1 + √17 and x = -1 - √17, which can be expressed concisely as x = -1 ± √17. We also compared our solutions with the given options and confirmed that the correct answer is B. This exercise demonstrates the power and versatility of algebraic techniques in solving quadratic equations. Understanding the underlying principles and methods allows us to tackle a wide range of problems effectively. The process of solving quadratic equations involves several key steps, including rewriting the equation in standard form, applying appropriate methods, simplifying expressions, and comparing solutions. Each step requires careful attention to detail and a solid understanding of algebraic concepts. By mastering these techniques, one can confidently approach and solve quadratic equations in various contexts, whether in academic settings or real-world applications. Furthermore, the ability to solve quadratic equations is a fundamental skill in mathematics and serves as a building block for more advanced topics. The exploration of different methods, such as completing the square and using the quadratic formula, provides a deeper understanding of the mathematical principles involved and enhances problem-solving abilities. The successful resolution of the equation x² + 2x + 1 = 17 underscores the importance of these skills and their applicability in mathematical problem-solving.