Solving Algebraic Equations Using Identities: x - 1/x 5 and x2 - 1/x2

Algebra is a powerful tool for solving complex equations. In this article, we will explore how to use algebraic identities to solve problems involving expressions like x - 1/x 5. We will specifically focus on finding the value of x2 - 1/x2 given that x - 1/x 5.

Introduction to Algebraic Identities

Algebraic identities are essential in solving complex equations. They are algebraic equations that hold true regardless of the values of the variables involved. By using these identities, we can manipulate expressions and simplify problems, making them easier to solve.

Problem Statement

We are given the equation:

x - 1/x 5

We need to find the value of:

x2 - 1/x2

Step-by-Step Solution

To solve for x2 - 1/x2, we can use a well-known algebraic identity:

x2 - 1/x2 (x - 1/x) 2 - 2

This identity is derived from the binomial expansion and is often used to solve such problems. Let's proceed step-by-step.

Step 1: Square the Given Expression

Squaring both sides of the given equation x - 1/x 5, we get:

(x - 1/x)2 52

This simplifies to:

x2 - 2 * (x * 1/x) 1/x2 25

Simplifying further:

x2 - 2 1/x2 25

Step 2: Isolate x2 - 1/x2

Now, we can rearrange the equation to isolate x2 - 1/x2:

x2 1/x2 - 2 25

This simplifies to:

x2 1/x2 27

Now, we use the identity x2 - 1/x2 x2 1/x2 - 2.

Step 3: Apply the Identity

Substituting the value we found:

x2 - 1/x2 27 - 2

This simplifies to:

x2 - 1/x2 25

Therefore, the value of x2 - 1/x2 is 25.

Alternative Methods

There are alternative methods to solve this problem. Let's explore a few:

Method 1: Direct Calculation

We can directly calculate the value of x from the given equation x - 1/x 5 by solving a quadratic equation:

x2 - 5x - 1 0

Using the quadratic formula:

x (5 ± √(25 4)) / 2

This gives us two solutions:

x1 (5 √29) / 2 and x2 (5 - √29) / 2

We can then calculate x2 - 1/x2 for each of these solutions:

x12 - 1/x12 26.96291201783626

x22 - 1/x22 -26.96291201783626

Method 2: Using the Identity Directly

Another way to solve this problem is by directly using the identity:

x2 - 1/x2 (x - 1/x)2 - 2

Substituting x - 1/x 5:

x2 - 1/x2 52 - 2 25 - 2 23

Thus, the value of x2 - 1/x2 is 23.

Conclusion

In conclusion, we have explored different methods to solve the problem of finding x2 - 1/x2 given that x - 1/x 5. Using algebraic identities and quadratic equations, we were able to find the correct solution. The key is to recognize and apply the right identities, which simplify the problem significantly.

Understanding algebraic identities and their applications is crucial in solving a wide range of complex mathematical problems. Whether you are a student or a professional, mastering these techniques can greatly enhance your problem-solving skills.

Keywords: Algebraic Identities, Solving Equations, Mathematical Solutions