Solving (x+2)^2 - 2(x+2) - 15 = 0 With U-Substitution A Step-by-Step Guide

In mathematics, solving equations is a fundamental skill. Quadratic equations, in particular, appear in various applications, from physics to engineering to economics. Often, quadratic equations are presented in a form that requires some manipulation before standard solution methods can be applied. One such technique is u-substitution, which simplifies complex equations by introducing a new variable. This article will explore how to solve the quadratic equation (x+2)^2 - 2(x+2) - 15 = 0 using u-substitution. We will break down each step, providing a clear and comprehensive guide for anyone looking to master this technique.

Before diving into the solution, it's essential to understand what quadratic equations are and why substitution can be a valuable tool. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations can be solved using various methods, including factoring, completing the square, or the quadratic formula. However, when faced with an equation like (x+2)^2 - 2(x+2) - 15 = 0, directly applying these methods can be cumbersome. This is where substitution comes in handy.

Substitution is a technique used to simplify equations by replacing a complex expression with a single variable. In this case, the expression (x+2) appears multiple times, making it an ideal candidate for substitution. By letting u = (x+2), we can transform the original equation into a simpler quadratic equation in terms of u. This makes the equation easier to solve, and once we find the values of u, we can substitute back to find the values of x. This method not only simplifies the algebraic manipulation but also provides a clearer pathway to the solution. Mastering substitution techniques is crucial for tackling more complex mathematical problems and is a valuable skill in various fields that rely on mathematical modeling.

To effectively solve the equation (x+2)^2 - 2(x+2) - 15 = 0, let's walk through the process step-by-step using u-substitution. This method simplifies the equation, making it easier to solve. By breaking down each step, we ensure a clear and understandable solution process.

1. Identify the Repeating Expression: First, observe the given equation: (x+2)^2 - 2(x+2) - 15 = 0. Notice that the expression (x+2) appears twice. This repetition suggests that u-substitution can be an efficient method to simplify the equation. Identifying such repeating expressions is the first crucial step in applying this technique. By recognizing patterns, we can transform complex equations into simpler forms, making them more manageable to solve. This skill is not only useful in solving quadratic equations but also in various areas of mathematics where algebraic manipulation is required.

2. Introduce the Substitution: Let's substitute u for the repeating expression (x+2). This means we set u = x + 2. Now, replace every instance of (x+2) in the original equation with u. The equation transforms into a simpler quadratic equation in terms of u: u^2 - 2u - 15 = 0. This substitution significantly simplifies the equation, making it easier to handle. Introducing a new variable like u is a common technique in mathematics to reduce complexity and make equations more solvable. This step highlights the power of substitution in transforming problems into more familiar forms.

3. Solve the Quadratic Equation in Terms of u: Now we have a standard quadratic equation in terms of u: u^2 - 2u - 15 = 0. This equation can be solved using several methods, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We look for two numbers that multiply to -15 and add up to -2. These numbers are -5 and 3. Therefore, we can factor the quadratic equation as (u - 5)(u + 3) = 0. Setting each factor equal to zero gives us the solutions for u: u - 5 = 0 implies u = 5, and u + 3 = 0 implies u = -3. Thus, we have found the two possible values for u. This step demonstrates the importance of being familiar with different methods for solving quadratic equations, as choosing the most efficient method can save time and effort.

4. Substitute Back to Find x: We have found the values of u, but our goal is to find the values of x. Recall that we defined u = x + 2. Now, substitute the values of u back into this equation to solve for x. When u = 5, we have 5 = x + 2. Subtracting 2 from both sides gives x = 3. When u = -3, we have -3 = x + 2. Subtracting 2 from both sides gives x = -5. Therefore, the solutions for x are x = 3 and x = -5. This step is crucial as it connects the simplified variable u back to the original variable x, providing the solutions in terms of the original problem. The ability to reverse the substitution is a key aspect of this technique.

5. Verify the Solutions: To ensure the accuracy of our solutions, it is always a good practice to verify them by substituting them back into the original equation. Let's substitute x = 3 into the original equation: (3+2)^2 - 2(3+2) - 15 = 0 simplifies to 25 - 10 - 15 = 0, which is true. Now, let's substitute x = -5 into the original equation: (-5+2)^2 - 2(-5+2) - 15 = 0 simplifies to 9 + 6 - 15 = 0, which is also true. Since both values satisfy the original equation, we have verified that our solutions are correct. Verification is an essential step in problem-solving, as it helps catch any potential errors and ensures that the final answer is accurate. This practice reinforces the understanding of the solution process and builds confidence in the results.

Having solved the equation (x+2)^2 - 2(x+2) - 15 = 0 using u-substitution, we found the solutions to be x = 3 and x = -5. Now, we need to identify which of the provided options matches our solutions. The options given were:

A. x = -7 and x = 1 B. x = -5 and x = 3 C. x = -3 and x = 5 D. x = -1 and x = 7

Comparing our solutions with the options, we can clearly see that option B, x = -5 and x = 3, matches our calculated solutions. Therefore, option B is the correct answer. This step highlights the importance of accurately comparing the derived solutions with the given options to ensure the correct answer is selected. It also reinforces the understanding that the solutions must satisfy the original equation, which was verified in the previous step.

When solving quadratic equations using u-substitution, several common mistakes can occur. Being aware of these pitfalls can help ensure accuracy and efficiency in problem-solving. One frequent error is forgetting to substitute back to find the values of the original variable, x. It's crucial to remember that solving for u is an intermediate step, and the final goal is to find the values of x. Another common mistake is incorrectly factoring the quadratic equation in terms of u. Double-checking the factors to ensure they multiply and add up to the correct coefficients is essential. Additionally, errors can arise during the algebraic manipulation, such as incorrectly applying the distributive property or combining like terms. Taking each step carefully and verifying the calculations can prevent these mistakes.

Another potential pitfall is misinterpreting the solutions. For instance, students might confuse the values of u with the values of x. Keeping track of the variables and their relationships is crucial. Furthermore, some may forget to verify the solutions by substituting them back into the original equation. Verification is a critical step to ensure the accuracy of the answers and catch any computational errors. By being mindful of these common mistakes and taking the time to check each step, students can improve their problem-solving skills and achieve accurate results. Emphasizing these points can significantly enhance understanding and reduce errors in solving quadratic equations using u-substitution.

In conclusion, solving the equation (x+2)^2 - 2(x+2) - 15 = 0 using u-substitution provides a clear and efficient method for tackling complex quadratic equations. By substituting u = x + 2, we transformed the original equation into a simpler quadratic form, which was easier to solve. We found the solutions for u, and then substituted back to find the solutions for x, which were x = 3 and x = -5. These solutions were verified to ensure accuracy, and we identified the correct answer among the given options.

Throughout this article, we emphasized the importance of understanding the underlying principles of u-substitution, recognizing repeating expressions, and accurately performing algebraic manipulations. We also highlighted common mistakes to avoid, such as forgetting to substitute back or incorrectly factoring the quadratic equation. By mastering these techniques and being mindful of potential errors, one can confidently solve a wide range of quadratic equations. The u-substitution method is not only a valuable tool in mathematics but also a testament to the power of simplification and strategic problem-solving. Understanding and applying such methods enhances mathematical proficiency and fosters a deeper appreciation for the elegance and efficiency of mathematical techniques.