Solving (x-3)^2 + 2(x-3) - 8 = 0 With U Substitution A Step-by-Step Guide

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In mathematics, solving equations is a fundamental skill. Quadratic equations, in particular, are encountered frequently in various mathematical contexts and real-world applications. This article delves into a specific method for solving quadratic equations: u-substitution. We'll break down the process step-by-step, using the equation (x-3)^2 + 2(x-3) - 8 = 0 as our example. By the end of this guide, you'll not only understand how to solve this specific equation but also grasp the broader concept of u-substitution and its applications.

Understanding Quadratic Equations

Before diving into the solution, it's crucial to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. These equations often arise in scenarios involving areas, projectiles, and various other physical phenomena. Solving quadratic equations involves finding the values of x that satisfy the equation, also known as the roots or solutions.

There are several methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, and, as we'll explore in this article, u-substitution. Each method has its advantages and is suitable for different types of quadratic equations. The choice of method often depends on the specific form of the equation and personal preference.

Introducing u-Substitution

The u-substitution method is a powerful technique for simplifying equations that have a composite structure. This method is particularly useful when dealing with equations where a certain expression appears repeatedly. By substituting a single variable, often u, for that expression, we can transform the original equation into a simpler form, typically a standard quadratic equation that is easier to solve. Once we find the solutions for u, we can then substitute back to find the solutions for the original variable, x in our case.

The beauty of u-substitution lies in its ability to make complex equations more manageable. It allows us to break down a complicated problem into smaller, more easily digestible steps. This not only simplifies the algebraic manipulation but also reduces the chances of making errors. In essence, u-substitution is a strategic simplification technique that can save time and effort when solving certain types of equations. It's a valuable tool in any mathematician's or student's toolkit.

Applying u-Substitution to (x-3)^2 + 2(x-3) - 8 = 0

Now, let's apply the u-substitution method to the equation (x-3)^2 + 2(x-3) - 8 = 0. The key observation here is that the expression (x-3) appears twice in the equation. This repetition suggests that u-substitution is an ideal approach.

Step 1: Identifying the Repeating Expression

The first step in u-substitution is to identify the repeating expression. In this case, it's clearly (x-3). This expression will be our candidate for substitution.

Step 2: Making the Substitution

Next, we introduce a new variable, u, and set it equal to the repeating expression. So, we let u = (x-3). This substitution transforms the original equation into a simpler form in terms of u. Replacing every instance of (x-3) with u, the equation becomes:

u^2 + 2u - 8 = 0

Notice how the equation has been transformed from a seemingly complex form to a standard quadratic equation. This is the essence of u-substitution – simplifying the equation to make it easier to solve.

Step 3: Solving the Quadratic Equation in u

Now we have a standard quadratic equation in u. We can solve this equation using various methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward approach.

We need to find two numbers that multiply to -8 and add up to 2. These numbers are 4 and -2. Therefore, we can factor the quadratic equation as follows:

(u + 4)(u - 2) = 0

This gives us two possible solutions for u:

  • u + 4 = 0 => u = -4
  • u - 2 = 0 => u = 2

So, we have found the solutions for u: u = -4 and u = 2. However, our original equation was in terms of x, so we need to substitute back to find the values of x.

Step 4: Substituting Back to Solve for x

This is the crucial final step where we reverse the substitution to find the solutions for x. Recall that we defined u = (x-3). Now we substitute the values of u we found back into this equation:

  • For u = -4: (x - 3) = -4
  • For u = 2: (x - 3) = 2

Solving these simple linear equations for x, we get:

  • x - 3 = -4 => x = -4 + 3 => x = -1
  • x - 3 = 2 => x = 2 + 3 => x = 5

Therefore, the solutions to the original equation (x-3)^2 + 2(x-3) - 8 = 0 are x = -1 and x = 5.

Step 5: Verifying the Solutions

It's always a good practice to verify the solutions by substituting them back into the original equation. This helps ensure that we haven't made any algebraic errors.

  • For x = -1: (-1 - 3)^2 + 2(-1 - 3) - 8 = (-4)^2 + 2(-4) - 8 = 16 - 8 - 8 = 0 (Correct)
  • For x = 5: (5 - 3)^2 + 2(5 - 3) - 8 = (2)^2 + 2(2) - 8 = 4 + 4 - 8 = 0 (Correct)

Both solutions satisfy the original equation, confirming that our solutions are correct.

Identifying the Correct Answer

Based on our calculations, the solutions to the equation (x-3)^2 + 2(x-3) - 8 = 0 are x = -1 and x = 5. This corresponds to option B in the given choices.

Advantages of u-Substitution

The u-substitution method offers several advantages, especially when dealing with equations that have repeated expressions or composite functions. Here are some key benefits:

  • Simplification: The primary advantage is the simplification of the equation. By replacing a complex expression with a single variable, the equation becomes more manageable and easier to work with. This is particularly helpful when the expression appears multiple times in the equation.
  • Reduced Complexity: U-substitution reduces the complexity of algebraic manipulations. It allows us to break down a complicated problem into smaller, more easily digestible steps, minimizing the chances of errors.
  • Clarity: The method provides a clearer pathway to the solution. By transforming the equation into a standard form, it becomes easier to apply familiar solution techniques, such as factoring or the quadratic formula.
  • Versatility: U-substitution is versatile and can be applied to a wide range of equations, including those involving trigonometric functions, exponential functions, and other types of composite functions.
  • Efficiency: In many cases, u-substitution can be more efficient than other methods, such as expanding and rearranging the equation. It can save time and effort, especially when dealing with complex equations.

Other Applications of u-Substitution

While we've focused on quadratic equations, u-substitution is a versatile technique with applications in various areas of mathematics, including:

  • Calculus: In calculus, u-substitution is a fundamental technique for integration. It allows us to simplify integrals involving composite functions, making them easier to evaluate.
  • Trigonometry: U-substitution can be used to solve trigonometric equations by substituting for trigonometric functions or expressions.
  • Differential Equations: In the study of differential equations, u-substitution is used to transform complex equations into simpler forms that can be solved using standard methods.
  • Algebra: Beyond quadratic equations, u-substitution can be applied to higher-degree polynomial equations and other algebraic equations with repeating expressions.

In essence, u-substitution is a powerful tool for simplifying mathematical problems across different branches of mathematics. Its ability to transform complex expressions into simpler forms makes it an indispensable technique for students and mathematicians alike.

Conclusion

In this comprehensive guide, we've explored the u-substitution method for solving quadratic equations. By applying this technique to the equation (x-3)^2 + 2(x-3) - 8 = 0, we demonstrated how u-substitution can simplify complex equations and make them easier to solve. We identified the solutions as x = -1 and x = 5, corresponding to option B. We also discussed the advantages of u-substitution and its broader applications in mathematics.

Mastering u-substitution is a valuable skill for anyone studying mathematics or working in related fields. It provides a systematic approach to simplifying equations and finding solutions efficiently. By understanding the principles and techniques outlined in this article, you'll be well-equipped to tackle a wide range of mathematical problems.

Remember, the key to success with u-substitution lies in identifying the repeating expression, making the appropriate substitution, solving the simplified equation, and then substituting back to find the solutions for the original variable. With practice and a solid understanding of the method, you'll be able to confidently apply u-substitution to solve a variety of mathematical challenges.